A. Sign in if you are not signed in already.
B. If signed in already, click the link Check for an updated license on the account settings page.
C. "We could not download a license. Please ensure your accounts are authenticated." If this message is encountered, sign out and sign back in. You can either sign out by clicking on the Sign out link and then signing back in by clicking on the account icon in the upper right corner of Visual Studio. Or you can just click the link Reenter your credentials. This link is a good indication that your account is not synced up with Microsoft servers and the error message is right in that your account is not authenticated and the license cannot be refreshed.
Once you have signed back in and your account is successfully authenticated, you will see that the "Reenter your credentials" link is gone. This is a good indicator that you can now go ahead and refresh the license. If it does not do that automatically you can click the Check for an updated license link once again. This time you will see a short confirmation message and your associated email address will be listed.
Signing in is inevitable if you want to get rid of this annoying warning. But even if you have always been signed from day one, you may still run into this warning. Based on my experience, this may happen if you have recently changed the password for your account, or you have added 2FA (two step verification) to your account, or you have reset the recovery code for the account and with it the app passwords. It may also have to do with inactivity or the recent changes Microsoft has been making to the "sign in experience" for Microsoft accounts.
Whatever the reason may be, signing in, or signing out and then signing back in again appears to be the best treatment for this annoyance.