There was an error connecting to the Apple ID server

You are sign in here .  .  .  with your Apple ID & Password (?).

Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.

You can’t merge multiple Apple ID accounts.

Check your Apple ID via your email box, see this Apple article: If you forgot your Apple ID

If you forgot your Apple ID email address or phone number - Apple Support                  

OR see -> Having trouble signing in?

https://iforgot.apple.com/password/verify/appleid

There was an error connecting to the Apple ID server

👉 See also -> If you can't connect or sign in to iCloud

If you can't connect or sign in to iCloud - Apple Support

See this Apple article: If you forgot your Apple ID password

If you forgot your Apple ID password - Apple Support

Then see this Apple article: What to do after you change your Apple ID or password

What to do after you change your Apple ID or password - Apple Support

See also the following Apple articles:

If you forgot your Mac login password

https://support.apple.com/en-us/HT202860

Reset your Mac login password

https://support.apple.com/en-mn/guide/mac-help/mh35902/mac

Change the login password on Mac

https://support.apple.com/en-gb/guide/mac-help/mchlp1550/mac

If you can't reset your Mac login password

https://support.apple.com/en-gb/HT212190

Ways to remember passwords on Mac

Ways to remember passwords on Mac – Apple Support (UK)

Sign in with your Apple ID

Sign in with your Apple ID - Apple Support                               

Sign in to your Apple ID preferences on Mac

Sign in to your Apple ID preferences on Mac – Apple Support (UK)

See if anything here helps.

User Password Reset.   -

User Password Reset (2).  -  

User Password Reset – Apple ID.  -

Reset user password - M1 chip computer

If you can't reset your Mac login password   *