All the ways I come up with (giving out keys) have the problem of how do you renew the key, and how do you cancel it, without knowing which is which.
e.g.
1. Connect to Mullvad over Tor, authenticate with real-world user ID
2. Use this to sign a blinded token
3. Use this to connect to Mullvad anonymously after some delay
The first run would be kind of dodgy, but after that you could get new session keys on a fixed schedule and switch them out at a random interval.
If they see that user A authenticates and 10 minutes later, key A comes online, that can be traced, but if you then wait a week, authorize key B, and then wait a few more days to start using it, you should be good.
In practice, this has way too many issues to work in practice. It still requires you to trust them not to e.g. log IPs and correlate it that way, so it's all just snake oil.