Actually Chopin received the French citizenship in 1835.
We also need to put this in the historical context.
At that time, French was used throughout Europe as the lingua franca, the diplomatic language, the official language of Russian, German and Scandinavian Courts. However while European aristocracy and elite were speaking French, less than a quarter of French people were able to speak French, as regional languages and dialects were spoken throughout the country.
So Chopin using a French name at that time, can be compared to today’s artists and bands taking on English names even though this is not their native language.