My GF has had similar experiences. We have had dudes come up and hit on her when I'm literally right next to her. I hate to be like an alpha douche and say something to make it clear I'm her boyfriend because what if this dude just is legitimately excited to talk to someone else about their favorite game. Similarly my GF does want to be "that girl" that says "I have a boyfriend" every time a dude tries to talk to her before knowing his intentions.

What we do now is she interrupts them early in the conversation saying "oh sorry hold on I need to ask my boyfriend something before I forget," turns to me "baby, do we need to pick up any milk?" And I usually say "yea we barely have any back at the apartment". This works but amazingly we still had a guy not take the hint and ask her on a date at which point I did step in and ask him "did nothing about the interactions between me and her make it clear that we are a couple? How sociall illiterate are you?" I'm sure he wrote a post about it on a fedora forum later that day.