Maybe you can use the concatenation of FirstName and LastName fields, no idea why Name is not available
($User.FirstName & ' ' & $User.LastName) != $Label.USER_MANAGER
But I would recommend you to use a Field to hold if a user is a manager
Here is a possible answer to your question:
If you are logged in as that user than I believe $User.Id should resolve to the logged in UserId. To reinforce this, if the logged in user can access dev Console, try running the following code in the Dev. Console execute anonymous:
See which Id is outputted to the debug log.