I own both the Air 2s and Mini 2. My question is can both drones be bound to a single controller? Obviously not trying to fly both at the same time but just to save space when traveling with both. Also, would it affect the DJI refresh by doing so?

Yes same controller will work on the Air 2S and Mini2.  I use one controller for my A2 and A2S to save space and weight.

As for DJI Care Refresh, not sure so maybe one of the DJI mods can answer. I'm guessing it won't affect it as long as you binded each drone to its own controller in the beginning when you first activated it. Afterwards using one controller to any activated drone should be ok.

Hi, djiuser_jZSX9yOUlmJ5. Thanks for reaching out! Yes, you can use a single remote for both your DJI Air 2S and DJI Mini 2. It shouldn't affect the DJI Care Refresh since it is bound to your DJI Account and to the Aircraft's Serial Number only.