There is no signup information when creating an account, and one of the payment options is "cash in an envelope". Even if they're lying and do keep logs, it's as disconnected from your identity as I think is possible (assuming you do use that payment option instead of the others).
It's also what Mozilla VPN runs on.
I was intrigued by this line in your post, and it turns out it's true. I'm not all that paranoid, but I kinda like that idea.
Also, right now there's a caveat about it:
"Cash payments are delayed due to corona
6 May 2020 NEWS
If sending cash is your preferred method for topping up your Mullvad VPN account, please plan ahead. The coronavirus is causing delays in postal delivery.
To avoid being stuck with no time on your account, send your payment well in advance. Mail coming from the US is taking four weeks longer than usual, and even post from countries in southern Europe and England are delayed."
Can anyone explain what the practical difference is between using Mozilla VPN and using Mullvad directly?