At one level this is a trivial question with an easy answer. A British person is a citizen of Great Britain, whose more formal and official name is the United Kingdom, or UK for short. That should be the end of that, and the recent vote vote by citizens of Great Britain (I think permanent residents may also have been allowed to vote) to leave the European Union, the "e," arguably reinforced the meaningfulness of this identity, especially in regard to the broader alternative of being a "European."
But then we have this problem that this vote appears to be stirring up divisions within these British people, with the Scottish in particular having voted strongly against the majority outcome, resulting in renewed pressure to have another referendum on Scottish independence, for them to cease to be citizens of Great Britain, arguably to cease to be "British." Is this identity then much more fragile than we might think it is? This gets pushed further in that people in Northern Ireland also voted to Remain, although not by as large a margin as did the Scots, 56% rather than 62%. The Welsh went with the national majority, indeed mirroring it closely at 52%, with the English making that the national average by more strongly supporting Leave and offsetting the Remain majorities in Scotland and Northern Ireland, with this even more strongly the case in more rural parts of England as London went strongly for Remain, almost as strongly as did Edinburgh and Glasgow in Scotland. There seem to be some pretty sharp divisions on this among the main identifiable sub-groups among the British.
This then suggests that we should take this internal division and apparent lack of agreed upon identity a bit more seriously. The word "British" comes from the word "Britain," which while often used as a short hand name for the entire nation, the United Kingdom, more specifically means the island of Britain, large island to the east of the island of Ireland. That the UK is "Great" Britain is partly because it involves more than just the people on that island, most notably the Northern Irish, as well as those on other much smaller islands such as Lewis (birthplace of Donald Trump's mother) where Scottish Gaelic is still spoken, and the Isle of Man, where the now-extinct language of Manx was spoken, related to Irish and Scottish Gaelic (and where the cats without tails come from),among some others. Thus being a British citizen includes people not living on the island of Britain.
The name "Britain" itself is quite old, going back to at least the Roman period, when those living on the island, or at least in the part of it ruled by the Romans, were known as the "Brythons." However, that proves to have involved a narrower group than those who live there now, not including the people now in Scotland who were called the "Picts" by the Romans, although they never viewed themselves as a group and identified themselves by tribal group names constituting sub-groups of themselves. That Scotland itself is sub-divided is clear in the division between the Highlands, where one is more likely to find people who can speak Gaelic, the language of people who invaded from Ireland several centuries after the Romans stopped ruling in the southern part of the island of Britain, the Romans having built Hadrian's Wall to protect the zone they ruled and full of Brythons largely to keep out the troublesome Picts, who reportedly painted themselves blue. Modern Scots are descended from ancient Pictish tribes, but also with this Irish Gaelic ancestry in the Highlands, as well as Viking ancestry, and Anglian ancestry in the Lowlands (Lowland Scottish fishermen reportedly can communicate easily with Frisian ones from the Netherlands, the Frisian language supposedly close to Old English). Yes, the Lowland Scots have serious English ancestry.
As for those original British, the "Brythons" who were ruled by the Romans, they lived in what is now England. But the language that the spoke was an ancestor to the modern Welsh language. Perhaps this is why the vote totals in Wales on Brexit so closely corresponded to the overall totals in Great Britain as a whole. However, clearly the modern Welsh are distinct from the Scottish, the Northern Irish (or "Scotch-Irish" as they are called in the US), not to mention the modern English. After all, Welsh is a Celtic language, if one more closely related to Breton spoken in northwestern France, as well as the now dead Cornish language, once spoken in Cornwall in the very southwestern most part of modern England, than to the Gaelic languages that came out of the island of Ireland. These modern Welsh are not all that closely related to the modern English, who now occupy the territory once occupied by the Roman-ruled Brythons, ancestors of the modern Welsh.
As it is, English not a Celtic language, although having many Celtic loan words in it,but mostly a Germanic one, related to modern Frisian as noted above, although also now with many loan words from Latin languages as a result of the Norman Conquest in 1066 and all that. The Germanic Anglo-Saxons (who also included the Jutes from Denmark who mostly ended up in Kent in the southeastern most part of England) invaded the island of Britain around and especially after the removal of Roman rule of what is now England and Wales, pushing the Celtic-speaking Welsh westwards into modern Wales, although also certainly killing many of them and intermarrying with some of their women to create the modern English. We indeed have some complicated migrations and wars that lie behind the identities of the main modern groups that inhabit both the island of Britain as well as the nation of Great Britain, not getting into all the groups that have arrived more recently ranging from Jews from Central Europe through Hindus from India and Muslims from Pakistan to Polish plumbers especially recently under the auspices of the European Union, from which the English in particular seem so keen on leaving, much more so than their fellow "British" in the Celtic fringe.
So we have it that the modern British are a bunch of sub-groups, ones that do not intermarry or mingle all that much, except maybe in London and a few other large cities. At some deeper level there really are not many "British" in Great Britain in the sense of people who are the descendants of fully intermarried members of these older constituent sub-groups who are very much aware of their identities, with this awareness if anything being heightened by their different attitudes towards this Brexit vote. This vote has if anything undermined what it means to be "British," even as it supposedly reinforces it. Indeed, quite a few observers are noting that this vote was really about the English asserting themselves, with those rural parts especially in the north and east often called "Little England" being the most strongly pro-Brexit parts of Great Britain of all.
While I have not seen anybody doing so, I am going to raise the question then about if there is a place where this intermingling of these different sub-groups has happened, where indeed we might find people who might represent this type that does not, or only barely does so in Great Britain itself. I think there is. It is the United States of Ameica, although also probably to a lesser degree in some of the other former English-speaking colonies of Great Britain, such as Canada, Australia, New Zealand, and parts of South Africa. However, in the US, this is not immediately obvious, and this is partly because most of these people are certainly not called "British" or even "British-Americans," but something else. They are called WASPs, or "White Anglo_Saxon Protestants," and I am one of them by ancestry, or so a sociologist who would use this term would argue.
There is a problem, however, with this label, which misleads most people to the real background of these so-called "WASPs," a term that was invented in the 1950s by sociologists and poltiical scientists, although sometimes the "W" in it is argued to stand for "wealthy," with the real WASPs being only the wealthier and more elite branch of this group, who arguably were long the dominant ruling elite of the US. The problem lies in the use of the term "Anglo-Saxon," which has the more specific meaning and association with the English of Great Britain. The term's specifically literal meaning is White English Protestants. But in fact only in certain parts of the US are the so-called WASPs largely of only English ancestry, especially in rural parts of New England (hence that name) as well as in the more Tidewater areas of the southeastern states, especially in Virginia. These people are more likely to be Episcopalian or Congregationalist or Quaker (or curiously Morman, with Utah probably the US state whose population is more strongly descended from the English than any other).
Most people in the US identified as being WASPs are of mixed ethnic descent. English is certainly a major part of it, but especially in the US South this descent usually includes people from Celtic fringe of Great Britain, the Scots, the Ulster or Northern Irish called the Scotch-Irish, especially in the Appalachian mountains, as well as the Welsh. My last name is Welsh, but I am descended from all these groups. And these WASPs often have other groups as well, mostly other Protestant northwestern Europeans, with the Dutch prominent in New York, the Germans in Pennsylvania, the French Huguenots in South Carolina, and the Scandinavians in the upper Midwest, as well as often some unacknowledged amounts of Native American, African-American, or others (I have both German as well as some Gypsy ancestry). And these people often adhered to religious groups not so strictly tied to the English as are the Episcopalians, Congregationalists, Quakers, and Mormons,such as Presbyterians (Scottish), Methodists (Welsh), Baptists (German), and Lutherans (German and Scandinavian).
So, the bottom line is that the real "British" are the British Americans now labelled as "WASPs." It was in America where this mixing of these groups that have not mixed so much back in Great Britain have mixed, creating that type that might have constituted a unified ethnic identity in the home country, but have not done so there. It has been in America where this mixing happened, even as the label applied to this group in the US suggests that it is mostly or only of English descent. In any case, whatever one thinks about it, the power of this group has been fading since the end of World War II.
I shall close this by simply noting that I because aware of this personally only about two decades ago, although I was intellectually aware of the fact that American WASPs, especially those in the US South (who include the lower class "rednecks"), were of this mixed English-Celtic ancestry. It was on a visit indeed to Great Britain when we went driving around, although I had done this more than once at earlier times. I kept realizing that I found myself sympathetic to and feeling a kindred with all of the people who were local to each area, even as I realized that I was not so fully sympathetic for the reason that I was not just English or Scottish or Welsh, but all of these in my ancestry. I realized that I was one of the "real" British, a British-American, somebody not found very often in Britain itself. Curiously this difference was long recognized between the British British and the British Americans, but in the earlier era, prior to World War II, this odd group that dominated in the US was simply called "Americans," although that term has now lost that meaning as it now means something like what "British" means in Great Britain, that is, somebody who is a citizen of the US whatever is their ethnic ancestry. Thus we have since identified that group with this oddly misleading term, White Anglo-Saxon Protestant, which is not precisely correct in general.