It seems like, perhaps, for trans princes and princesses, one could use die Prinzin and der Prinzess, but I am not certain that is the reason the words “Prinz” and “Prinzessin” developed as they did. Does anyone know ?
Like in English the etymology of both, the German Prinz, and Prinzessin is from old French prince and princess. These again are derived from Latin princeps (principle).
So the peculiar male and female forms did not develop in German but in French. Interestingly in German there was an interim word Prinzesse which is closer to the French root. The additional Germanized ending -in came later.
Well, if you start with the English terms
(and quite similar in French, which should be deemed the direct source of these words in German, as French was for formative centuries the lingua franca in higher society in Europe)
you get German forms
Now, German "Prinzess" (for a female such) is very unusual (although not unthinkable1); I would say it was just because there is a common practice to add -in to titles like this for female forms, the -in was added to princess, too, although in French/English the -ess would already be sufficient to indicate feminitiy.
1) See examples and historical evidence given by commenters below.