I have anxiety, my boyfriend doesn't. So yes.
But then again maybe mine is not as severe.

Yes, I think it's possible. In my case, I've had social anxiety for a long time and being with my first boyfriend made me more comfortable and outgoing. Being around big groups of people and going out in general was almost no problem with me. You should be each other's comfort zones or w/e and if one isn't trying and they're in a relationship, then I don't even know why they're in a relationship. Trying is the key.