Is it even possible for somebody to become intoxicated from using mouthwash?

Yes: If the mouthwash contains alcohol and it is swallowed.

If You Drink It: If you swish & spit, you'll be fine. You can, however, get drunk from mouthwash if it contains alcohol and you drink enough of it. Obviously, that's not recommended. If you have a history of substance abuse, you should absolutely not use an alcohol-containing mouthwash. You should also keep it away from children. Rarely, some people can't metabolize alcohol at all, so they need to be very careful.