I don't think I would have been embarrassed and wanted to say that the Doctor was my grandfather; I'd be happy to call him "a friend who's helping me with my stuff"... but then again, I'm not a twenty-something trying to make a good impression on my housemates.

It's not just embarrassment, it's super inappropriate for a senior lecturer like the Doctor to hang out with an undergrad like Bill, even if he is tutoring her. It would inspire gossip that would affect both of their reputations, but especially Bill's.