putting aside that the air is currently fine in CM, apologies, I misread what you said. The inverse is true. If you're living in Phuket, not CM, you should get the car and bikes registered there. You should have plates reflecting where you live.
If you don't mind me asking - why Phuket? I can understand why you'd hate March in CM but it's not a place I'd move to down south given a choice. Everything is twice as expensive and the locals aren't nearly as nice.
is not wrong - you can keep the vehicles registered in Phuket, if you're in Chiang Mai for the long term and don't want to keep traveling to Phuket (noting that vehicles over a certain age require an inspection) it's probably just easier to transfer the registration of the vehicles to your Chiang Mai address. You will have to change plates in that situation - with the exception of Bangkok plates your plates are meant to reflect where you are living. It's about 200 baht per vehicle to transfer the registration and plates, probably about the same, maybe less for motorbikes.
the nuts part is that the government, two weeks after it started spreading from Bangkok is that they haven't stopped inter-provincial travel. They should have shut it all down.
Any person entering Chiang Mai Province from Bangkok, Pathum Thani, Nonthaburi, Nakhon Pathom and Samut Prakan must do home quarantine for 14 days, register with CM Chana and report themselves to an epidemic control officer.