General Apple ID is say example1@gmail.com you created this mail address from web gmail server and then signed in Sign in with your Apple ID - Apple Support after that signed out apple ID and created a new Apple ID say example2@icloud.com it started working in Mac using previous account credentials How to create a new Apple ID - Apple Support
Then this Apple ID password was changed Change your Apple ID password - Apple Support
The general web mail password is still different on web server , at present the example2@icloud.com is signed in which the Apple ID password is changed .
Note : now example2@icloud.com is treated as Apple ID and functional , however the example1@gmail.com is acting like a web mail address , but it will be treated as apple id when you sign out example2@icloud.com and again sign in with example1@gmail.com