This is a tricky question, precisely because questions of identity are difficult to tackle, and how we label individuals in hindsight often clashes with how that same person might have understood themselves in the context in which they lived.

Honestly, I am not convinced that Richard would have called himself a Frenchman or an Englishman, these are identities more commonly associated with the rise of nationalism that just aren't a thing at this point in history. Certainly there was a concept of an Anglo-Saxon people and an understanding of a territorial realm under the nominal rule of a King of France, but there really weren't commonly held, unified national identities, especially in France. More than likely, Richard would have identified with his Angevin and Norman heritage. Granted, throughout the eleventh and twelfth centuries we see the gradual formation of an Anglo-Norman aristocracy in England- so this is definitely not a period of static identity, which makes sense; identity tends to be quite fluid.

I would hazard that we refer to Richard first and foremost as the King of England because people tend to care more about and are more invested in the office of kingship- not many folks are bothering to trace the lineage of the Dukes of Normandy. Which title Richard valued more (duke or king) I personally can't speak to, but in the general history that's led to the formation of English and French national identities, people seem more concerned about counting crowns and kings than dukes and counts. A kingdom was and is, generally, more prestigious than a dukedom.

On the subject though, it is interesting that among historians the recent trend has been to refer to the lands ruled by Henry II and his children as the Angevin Empire to better describe the multinational nature of the territories and people they ruled.