Messages – Can’t log in.
Messages – Can’t log in 2.
Messages - If you can‘t sign in to your iMessage account or FaceTime on your Mac.
Have someone else attempt to sign in using their Apple ID. If they can successfully sign in and use Messages/FaceTime, it may be your Apple ID. Someone I know had this happen and I was able to sign in using my Apple ID. After a call to Apple Support and then being elevated to a tier 2 support person, it turned out it was a problem with their Apple ID. It took the software engineers 3-4 days to fix the problem. When calling Apple, make sure you tell them it is an Apple ID problem so you don't get charged.
You can also try signing in on another device/computer. There have been a couple of posts where people's device/computer was blocked. You will need to contact Apple for that problem also.
There is a contact link.
Apple ID - Official Apple Support.
Contact Apple for help with Apple ID account security - Apple Support. (The link doesn’t always load)
Get Support Work through the screens to get a chat option.
Apple Support schedule phone call
https://twitter.com/AppleSupport
Are you running any anti - virus programs? Using a VPN?
Try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Also try the Safe Mode.
Isolating an issue by using another user account
Safe Mode - About.
If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor. Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply. Pastebin.
Etrecheck – System Information.
Web site version if required. https://etrecheck.com