This happens to a lot of people and when they get home and even after they finished bowling, there is a thing called bowler thumb which is a injury resulting from compression or repeated friction on the inside of the thumb which causes pressure on the Ulnar nerve and I have written a blog post about that here: https://www.rotogrip.org/bowlers-thumb-treatment/
But other reason why your thumb is sore after bowling is due to that you are squeezing the bowling ball to hard and that can make your thumb swell up and get sore. When rolling a bowling ball, the first finger that comes out is your thumb so if your thumb is sore then you are not rolling the bowling ball the right way and that can cause soreness too. Your thumb can swell up a bit while bowling but it should not be sore.
There are some things that you can do to help with your sore thumb. The first thing that you can do is use bowling tape. Bowling tape will help your bowling ball come off easier so you won’t have to squeeze the ball. If you need some bowling tape you can click here: Try not to squeeze the ball too much. Take your bowling ball to a pro shop and telling them what’s going on. Maybe your thumb hole in your bowling ball is to lose or to tight which may cause your thumb to be sore.