Do you think it's wrong for me to jokingly share my coworkers mistakes with my team?
Why would you share that?
Because he is so bossy, I have to point out his mistakes even if it's a minor one
You should just let it go and not tell everyone. We all make mistakes and that's really unprofessional to do something like that.
You're way too serious about things that aren't that important.
It's important to maintain a professional demeanor at work. Consider how you might feel if your coworker chooses to point out your mistakes in the future.