A Discord bot that connects to a local database (OOP)
Is this the correct way to access those methods and their variables in my register command? 


Yes. Looks good to me.

I can't say I'm fond of your docstrings. But your code is perfectly clear.

You are focused on MySQL / MariaDB. I tend to access mysql through sqlalchemy, just in case in future I'll want to use sqlite or postgresql or another.

The code starting if discord_id in ... is a bit odd, perhaps it could go into a function? It is a usual expectation that calling code could import your module (define your class) more than once without odd side effects.