Why is it both "Mai" and "My" used for the title?
There doesn't seem to be any official word on why both romanizations are used. However, Funimation's official name for the show is My-Hime, while the technical romanization of 舞 is mai. As such, it seems as though sites use Mai-Hime because that is the technically correct romanization of "舞 Hime" and use My-Hime because that's the title that companies such as Funimation and Bandai use for the show (just as how Attack on Titan is also listed as Shingeki no Kyojin).

EDIT: to add/expand on what Aki Tanaka mentioned from Japanese Wikipedia, it seems the name is also a pun, where because it's about moe characters, the creator wanted people to "please find (My) princess (HiME) for oneself" (approximate translation, because my Japanese is rusty and direct translations are hard). HiME is also a backronym for Highly-advanced Materializing Equipment, which is the ability that the main characters have.