You can put the rebate in another name as long as they are on the registration.
I went though this with my latest Model 3.
My car has both my wife and i on the Registration. Since i wanted her on the registration, my credit union required her to be added as a name on the loan, but her credit was never checked, and the loan was not added to her credit report (So she technically didn't need to be on the loan in the eyes of CVRP.
The California rebate was applied in her name only. All they asked for in terms of ID was her California drivers license.
I happened to use a different address because we moved since our last Model 3, but the address doesn't really matter. The only instance that it would matter is if you sold your previous car that you got a rebate for before the 30 month required ownership period. It would be a way for them to catch you on that.
We plan to get her a model Y next year. She should have 1 more rebate available to her as long as the funds to run out.
Hope that helps!