Factory resetting MacBook asking for password on an ID that it says does not exist

You have not followed the steps for selling a Mac. See What to do before you sell, give away, or trade in your Mac and Factory reset of your Mac - Apple Support.

If the computer is a Late-2011 or later model, then do the following:

Internet/Network Recovery of El Capitan or Later on a Clean Disk

1. Restart the computer. Immediately, at or before the chime, hold down the (Command-Option-Shift-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the target drive (out-dented entry w/type and size) from the side list.
5. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
6. Set the partition scheme to GUID.
7. Set the Format type to APFS (SSDs, only if installing Mojave or Catalina ) or Mac OS Extended, (Journaled.)
8. Provide a volume name, usually "Macintosh HD."
9. Click on the Apply button, then click on the Done button when it activates.
10. Quit Disk Utility and return to the Utility Menu.
11. Select Install OS X and click on the Continue button.

In that case, you should contact Apple. I can't help you with an Apple ID issue like this.

Contacting Apple Customer and Support Service

1. Apple Store Customer Service at 1-800-676-2775 or visit online Help for more information.
2. Primary support help at Contact Support.
3. To contact product and tech support:  Contacting Apple for support and service including international calling numbers.
4. Starting  an  online  chat  with  Apple  Support.
5. Apple  Support  using  Twitter