$date variables in body text

I can tweak the predefined system variable Modification Date Long to end up with a variable that shows {monthname year} in body text

Since the system variable can be customized and is working in your documentation, I think I'm missing why you would prefer to use a user variable instead. The idea behind user variables to allow us to create variables that Adobe didn't think to provide for us in advance. But both are true variables, inserted, updated, and imported across docs the same way.

And to the best of my knowledge, no, we still can't add date and time building blocks to user variables, even in 2015.