And Ireland, of course, does not consider itself British. Ireland was completely overcome only a few times, despite the Medieval depredations of the terrifying Norsemen. The Romans scarcely even tried that island. Well, Scotland: the reason the Romans built that wall was the Scots wouldnt bend the knee. Thats the Gaels for you. Sure, the British eventually won over Ireland, but the Irish decisively never gave in.
But look at the English language itself: English is closely related to German, but retains many words from everybody else. Angles, Saxons, Jutes, Frisians, Normans: they came, they conquered, they were absorbed.
I visited London once, and thought to myself that the former Empire was fully represented on every street. Regardless, people born and raised in the UK are British and consider themselves British very much as we Americans do.
I studied another island nation in college: Japan. Now that is a nation concerned with its cultural and racial purity. For centuries they isolated themselves. A couple of times when it looked like they were about to be invaded, a typhoon arose and sank the enemy ships the Divine Wind, or Kami Kaze. Revere the Emperor, expel the foreigner was a slogan of long duration.
The British Isles, otoh, has no such purity in the first place. The DNA of the the Royals of Europe criss-crosses throughout Northern Europe, Western Europe, and Russia as marriages were arranged and alliances sealed.
Im surprised at you, actually, to think that a DNA test would make any difference. Its all European, and various books, archives, and genealogies make titles and lineage publicly available for research. The only thing theyve never had to deal with was someone who was, all three, a foreigner, a commoner, and of mixed race. That upset their little applecart.