What to do if I can't cuddle because it makes it hard for me to breathe, is it asthma or claustrophobia?

The squeeze. If you have trouble breathing from a bear hug , that's normal. If you had trouble breathing from the hug, you have a social anxiety issue.
Three possibilities. If you have asthma and it is triggered by sexual situations, get it treated. Probably there is something else that triggers it. If he's holding you too tight, and it's not comfortable, tell him nicely ("I LOVE it when you hold me -- just let me breathe"). Some people simply don't like being held, even by someone with whom they're in love, and this can make you feel your breathing. Is this you?
Not asthma. This is probably not asthma, unless you are sensitized to something the other person is wearing (perfume, or moldy clothes). If it goes away immediately when you stop, it's probably not asthma. Any breathing concern should be evaluated by your doctor, who can help you sort that out.