It seems likely that when she served as Princess of Theed (basically, governor) and then ran for and was elected Queen, she used the name Amidala. Even on Naboo, it would be very strange to elect someone named, say, Mary, only to end up with an elected official named Elizabeth.
Padmé has three names: Amidala, the aforesaid Padmé, and Naberrie. Going just by the movies, it's not entirely clear what role each name plays, but given that her other family members use Naberrie, that seems to be her birth family name (surname). She uses Amidala when she is queen, but also when she is a senator, so it seems to behave as an assumed surname (in "Senator So-and-So", the "So-and-so" part is usually the senator's last name). Note that this is different from a regnal name, which would generally be a first name.
So, whatever role Naberrie and Amidala play, it seems clear that Padmé is her first name, and is what her closest friends always call her, regardless of what other name she's currently using. Given the similarity of the other handmaiden's names, Sabé, Rabé, Dormé, etc. are all also given names.
Thus, your question boils down to, why did Padmé continue to use her own given name when she was in disguise?
- Why she didn't use Sabé instead is clear: a handmaiden is much less disguised than the queen (there's a certain lack of face paint), so anyone who knew Sabé would know immediately that something was screwy if Padmé introduced herself as Sabé.
- Why didn't she invent another given name for her handmaiden persona? This is getting heavily into supposition, but I think it's because Padmé is a common given name. If Naboo was the US in the 80's, it would have been entirely believable to have both the queen and a handmaiden named Jennifer; Padmé doesn't seem to be quite that bad, but still, there are only so many names you can come up with that end in -é.