No, I’d definitely feel annoyed in your circumstances. You asked him not to say anything, he broke his promise. You’re justified in feeling annoyed by that. It’s understandable you worry about being judged for having mental health issues, but unfortunately you can’t control what other people think.
His mum either misinterpreted what you said, or he relayed it to sound like you’d meant it. Either way, it is super annoying when people take jokes seriously, though I can understand why she might have.
To answer the question of what I’d do, I’d probably try and make this friendship more casual, if I really wanted to hold onto it. I don’t think his mum has really done anything wrong, but it would bother me that she knew things I told my friend to keep quiet about.