Hello again!
Authenticating your account doesn’t necessarily have anything to do with the “Authenticator”-token - that is just a similarity in name, but it is obviously a very good idea to have an Authenticator ij place to protect the security of the account.
The error you’re describing here refers to your computer not being able to create a connection towards the battle.net servers, which prevents the games from being recognized/enabled on your account. Typically this would be the result of an overly aggressive security program or incorrect network setting getting in the way - you can find instructions on how to deal with the most common issues of this nature under this link.