If your Apple ID is say example2@icloud.com but not signed in computer will be treated as an Apple ID , you might have signed a different Apple ID say used with gmail services .
Note : in the initial stage when Mac / computer is set up its a thumb rule you can't create directly an Apple ID with iCloud domain , you have to take help from third party mail services like gmail , yahoo ...to create an Apple ID .
It depends upon the user to sign out primary Apple ID say example1@gmail.com and click on create a new Apple ID the Apple ID say example2@icloud.com , so in the upcoming steps the account credentials like three answers for respective questions , date of birth is required ( for secondary authentication account ) , the steps will vary for 2FA account , so both id's are interconnected , however this mail address will be always treated as an Apple ID .
It depends upon you to reset the password if forgotten from this link ( using a web browser ) https://iforgot.apple.com/appleid#!§ion=appleid
You know the Apple ID password for example2@icloud.com so it can be once again changed and not reset .