I’m not sure. If you look at the other Disney princesses, for the most part they are either born royal or marry royalty. Even Moana and Pocahontas are the daughters of chiefs, which would be royalty of a sort. Since Hercules gives up his “godliness” to stay with her, technically he could no longer be considered “royal”, so therefore by staying with him she does not gain any type of royal status.
However, if the sole criteria for being a Disney princess is that by the end of the movie the heroine has proved herself to be a good person who puts others first, then she would qualify due to the changes her character makes.