I still think it's worth a conversation with her. Mort is asking how to handle this, so it clearly isn't that clear cut for him.
Where's the harm in asking her why she doesn't want mutual friends to think they are bf and gf? She could have been testing him to see if he said "Why, we are bf and gf?!".
Women do weird things like that.
If she says "because I don't want to be your gf" or "I don't think of you that way" then he can go off and date the other woman with everyone knowing where they stand!
