If a person gets that feeling after any exercise, yes, it's not a bad idea to get checked for asthma. But if it only happens after swimming, I would think it much more likely that the issue is related to that pool.
Chlorine allergy is also a possibility, if the person experiences symptoms in pools that don't occur elsewhere.
I've also done some high intensity training outside of water without the same lung issues. I'll get out of breath after running for a while, but the pool related issue is every time I breathe in deep to expand my lungs, it would feel very strange and hurt. It would last even an hour after leaving the pool area.