I think it’s precisely because Picard is uncomfortable with that sort of thing, and probably has a reputation for it. I hear that hint of “unamused” too, but I think it’s at Picard’s immaturity with respect to the situation rather than “Why would a starship captain allow this to happen”.

I’m not sure the ribbing is that gentle, though I don’t read it as aggressive, either. More of a “Captain Picard Day? You’re actually participating in this, Jean Luc?” I don’t think they’re that close, probably only by work meeting/reputation.

I think either reading could be accurate, though. I don’t think this Admiral appears in any other episodes so we don’t have a broader picture of her.