When Mike and Vanessa adopted Bloom,how did they know that her name was Bloom?

I don't think they really did know that her actual name was Bloom. Since Mike found her during a 火災, 火 emergency, the only way of telling that she wasn't from Earth, was she was surrounded によって that sphere and Daphne was calling to Mike asking her to "save" and "help" and "take care of me". (It never gave any indication that the little girl's name was Bloom)

In my personal opinion, I think the two adults just chose the name and coincidentally, it turned out to be her actual name. (I think the creators did it so we wouldn't have to get used to another name since she sees 画像 of her parents throughout the series)

She had a ネックレス that have her families name on it.

I think that she might have a ネックレス または something on her Blanket that says her name