They both went by the same 'first name' because it was their family name.
If this is the sort of thing the OP means, then I think it's fair enough for her to request that. I was thinking that she was insisting on being called Mrs/Miss/Ms Whatever, which sounds superior and up herself (even if she isn't, that's how it comes across).
Otherwise if say, she's Mrs X and wants to be called X, rather than by her forename of Y, then I think her PA should respect that.
And TBH, if someone insisted on calling me by a name I considered personal (to family) or disliked (I've known many people who dislike their given name, some of who ask to be called something else, and I'm fine with that), then I would insist on calling said PA by some random name that wasn't her own. To ignore someone's wishes in this situation IS disrespectful to them.
If this is what is happening OP I can understand you being upset by it. It's almost as though your PA were calling you "Darling" or "Sweetheart", if use of your first name is an intimacy reserved for immediate family, and I certainly would make a point of telling her so (as well as the gossip thing - that's well out of bounds.). Here is doesn't matter how lovely a name is, it's a private thing and should not be used beyond the family.
SO:
If OP is insisting on (title + surname) in an office culture which uses forename - she is being U
If she insists on (surname only, rather than forename) in the same office culture, she is not being U, because that is her preferred name, in the same way as Thomas may prefer Tom, or Ethelberta Diane may choose to be Diane.
The gossip, whatever the circumstances, or even the innocence of it, needs to stop because it is none of the PA's business (e.g. a hospital appointment, if anyone needs to know where the OP is, is just an appointment - it's not "She's gone to get a boil on her bum lanced"). Also, being a PA means that you are expected to respect confidentiality.