In your terms their is nothing giving u permission to search my password for a breach in fact you should have no access to my password information. I moderate several forums much bigger than this and this is not common practice. Yes you can monitor a members email/username for issues within this site only, you do not have permission to use / monitor it outside of the site that is the user's responcibility only and anything outside of this site does not effect the site. You are now crossing a fine line of privacy... so now I wonder if u are accessing our passwords did the data breach come from here if so...
I think you're reading into this.
If I'm taking
@todo10 's post at face value, it's your email address that's being checked, not your password.
Forums get a crapton of spam, bots signing up, etc.
They likely subscribe to a service that checks addresses for compromise.
You had over your email address to a forum. (disassociated with your name, phone, or other PII because it's only a forum)
They likely received spam from your account or are just being proactive and upload the entire user list of email addresses to their service.
You're flagged.
In order to protect their forum, they turn on MFA for you to make sure you're not a bot.
This is not a big deal.
What is a big deal is that you've been compromised. You should think about using MFA on EVERYTHING, changing your passwords, use a password manager, and use unique passwords everywhere. Your cyber hygiene is lacking, apparently.