Why is it both "Mai" and "My" used for the title? When we look at the title for Mai-HiME and Mai-Otome it's written in 2 ways


Mai-HiME/Otome
My-HiME/Otome


We can see this in Manga Updates, My Anime List, Anime Planet and Wikipedia where either they list the alternate names showing both Mai and My to making a choice between My or Mai.

When we look at the list of Associated names on Manga Updates it shows


舞-HiME
Dancing Princess
Mai HiME
My HiME
My Princess
My-HiME
Princess Mai


considering that one of the characters (the main one in Mai-HiME) is Tokiha Mai and the Japanese used for the title is 舞-HiME with google translate showing 舞 is pronounced as "mai" which it translates to "dance", the associated names have different 3 different meanings


Possession of a Princess/Hime
Referring to Mai as a Princess/Hime
matching the google translation of 舞 (mai)


I am wondering why does sites associate "My" and "Mai" for the title? to me it seems both "My" and "Mai" give the title different meanings
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.