Addressing save the date when dating...?

Yeah, if you’re inviting the two of them, not just her and whoever she decides to bring, list both names. You aren’t defining the relationship, just who is invited to the wedding.

If they stop seeing each other, are you fine with her bringing someone else as her plus one? On the other hand, if you list him by name, are you fine with him still attending your wedding if they have a falling out?

I’d do a plus one. I assume you’re sending to her with both names or a plus one, not individual invitations to each of them. He’s being invited as her guest, so if they stop seeing each other, I’d assume he is no longer invited. If he is, then I’d do a completely separate invite to him specifically at his home address.

Two of my fiancé's friends have a similar situation. One of them dates multiple women, the other is on and off (officially, not officially ??) with another girl. So when I send the invites, those people who I don't know what the situation is, will get a Mr. John Smith & Guest. I think that's the safest way.

If it’s just the save the date you can just address it to her. I didn’t indicate “guests” on my stds, unless they were couple friends of ours and both would be invited regardless. You can also always just reach out to her and be like “hey I’m sending this to you but obviously he will be invited as well”