I think this is mo different than the general question of privacy on lightning. The payor of course knows who they are paying amd the payee knows who they are recieving from. But since routing nodes can only see the peers on either side of them, they don't know who is the payee and who is the payor. They effectively have no insight into who is the UBO.
Discussion
And taking this further... with braiins they are nonkyc to begin with (aside from email address) so they can't tie your recieve address to a proper identity unless that email or your IP gets doxxed. But those are problems not unique to lightning per se.