Mac os x lion reinstall error

You must also include the Two Factor Authentication Code as well when entering the password. Enter the password, but don't submit it yet. Immediately add the 2FA code, then submit the password & code together.

Try booting into Recovery Mode using Command + Option + R instead which theoretically should not require authentication for reinstalling macOS (or it may access a later online macOS installer if the Mac has access to Internet Recovery Mode).

If you can still boot macOS normally (or in Safe Mode), then you can get the exact model by clicking the Apple menu and selecting "About This Mac".

How old is your Mac?

http://checkcoverage.apple.com/ can help you identify the age of the Mac using its serial number. Do NOT post the serial number here. Just the age, that will tell us more about what is possible.

Keep in mind Lion is 6 years older than the operating system that this forum you posted to is about. What are you running currently? If you are trying to downgrade, the following link may be helpful:

How do I downgrade my Mac OS X operating … - Apple Community