I'm a medievalist specializing in 12th C England and France, so I'm sure other people do this differently depending on their area of interest, so this is by no means a definitive answer.

I typically identify historical figures by the cultural/ethnic identities that they match with the cultural, political and ethnic divides that existed at the time. For example, I would not call Henry II or Eleanor of Aquitaine 'French', because while they were both born and spent much of their lives in what is now present-day France, they were from territories that, in the 1100s, were vastly different from each other. I would call Henry II "Angevin", which is the term for a person coming from the County of Anjou, though he also had decidedly Norman influences, and I would call Eleanor of Aquitaine "Poitevin", as she was born and raised in the county of Poitou, and heavily influenced by the culture of Southwestern France, which was very different from the Angevin-Norman spheres of influence, and the Frankish culture of the Capetian dominions.

It is relevant to consider how people divided themselves throughout history, especially when we don't necessarily have the same conventions. It really bothers me when people point out that Richard I "didn't speak any English" as a negative point towards his Kingship when, in reality, virtually none of the nobility in England at the time would have spoken English (or rather, Anglo-Saxon) or would have had any interest in doing so. In fact, referring to them as English or Anglo-Saxon would have been as an insult.

There is currently some contention going on within academic communities regarding the use of certain terms, including 'Anglo-Saxon', as some terms have been co-opted by racist entities, which changes their. This is something that has not affected me so directly, as I'm focused pretty heavily on Norman and Angevin studies, but is definitely something I would need to start taking into consideration if I were to start writing anything for actual publication.

It is also important to remain consistent with current scholarship, and with common labeling given in primary sources in order to reduce confusion.

Edit: for clarification, as my period/region of study is fairly specific, and it was brought to my attention that I hadn't specified.

I would say that there is no real golden formula for this, but that there are actually two things to consider, and that's both a historical figure's self-identification, and also now a community identifies with a historic figure.

I can give a couple examples to show what I'm talking about. Take the Lithuanian painter and composer Mikalojus Konstantinas Čiurlionis. Now, if you look through his history, you might actually question just how "Lithuanian" he was - he was born in Vilna Governate in the Russian Empire, his family was Polish-speaking, he studied in Warsaw, and in fact he didn't even learn Lithuanian until his early 20s. Furthermore, he died in 1911, which was years before there was even an attempt at an independent Lithuanian state. But - he is considered Lithuania's national artist, and considered himself Lithuanian, and played a major role in the development of a Lithuanian national culture and identity.

Now of course people will say "well, he was always Lithuanian, and just rediscovered his true heritage", which is true from a certain point of view, I guess, although it tends towards a view in nationalism that there are objective ur-ethnicities or nationalities as opposed to shifting or changing national identities. The latter is historically a more accurate point of view, but Čiurlionis in particular is interesting because he's a prominent example of someone who arguably undertook that shift personally in their own adult life.

Now for another example to show my second point, namely that a nation or group can associate with an historic figure. And here I'm thinking of Al-Farabi, a 10th century philosopher and scientist. Al-Farabi has been invoked by the modern state of Kazakhstan - there is a national university named after him, and his (supposed) likeness graced the country's money. Now precious little is actually known about Al-Farabi's life, let alone how he identified himself - he may have been born in Otrar, in present-day Kazakhstan, but he might have been born somewhere else in Central Asia (the most we know is that he was from "Khurasan", which was very vague in its definition at the time). He probably spoke Persian, and wrote in Arabic, and spent most of his life writing and studying in Baghdad. Was he Kazakh? Absolutely not - no one identified as such for a good five centuries after his time. Was he Kazakhstani? Same, no such state existed at the time. But do modern people in Kazakhstan identify him as an important figure in their patrimony? Absolutely. So in this case he is identified with a nation and a nationality that he wouldn't have claimed himself. I would say something like this is similar to King Arthur and England.