Messages keeps asking for Password

You are in an administrative user account, right?

Check the Sharing and Permissions section of the Info pane for the Messages app to see if you have Read and Write permission. If not add yourself with Read and Write permission.

If the above doesn't help give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

I was looking at a 3rd party app and to an Apple app. The permissions for my Messages app are:

So a custom is OK.

Create and log into a new, basic, admin user account, Set up users, guests, and groups on Mac,  and see if the problem persists. This tells us if the problem is limited to your user account or is system wide.

Note:  do not launch any 3rd party apps in the account so as to not install any of the supporting files that most create.

If the problem isn't present then your original account has is the culprit and more investigating needs to be done.

See if you can add yourself with read and write permissions.