itch.io

Sounds like a scam. Especially if they lure you with their own connections.

Or is this an actual reputeable university and the enroll price of one of those countries where education costs a fortune?

Maybe start with remote courses that are meant for people that have a job. Or better, do some jam games for free and fun. You can code. You will quickly learn, if game making is even something you wanna do. Apart from story, a game is nothing more than a complex algorithm. Making this fun and rewarding is game design. But coding it in the first place, would be a skill you should already have.

I think there is a big difference between a specialist coder working for a game studio and a solo developer or tiny studio. Solo developers very often do their own art and basically everything. Creating a game is such a big field of individual skills you can apply. Writing. Game design. Graphics. Classic AI. Level design. And so on. Including marketing.