How do I recover Game Center password if it's different then my Apple ID password?
This is a common confusion point.

All apple accounts are AppleID and you can/should first go try to reset or recover the password for each account at https://iforgot.apple.com

Any use of Apple accounts on iOS (messages, FaceTime, game center, app store, iCloud) are necessarily first AppleID accounts - so mixing several of these on one device can cause a little confusion on which password belongs to which.

There is no Game Center account or password retrieval (nor any other account) other than the main iforgot page. You can go in and check your AppleID and learn more at https://appleid.apple.com