"Residence" for what purpose? Customs? Drivers' license, vehicle registration and insurance? "Residence" is an imprecise legal concept with varying meanings and definitions depending on context. Sometimes its fixed by statute, sometimes case law and sometimes by nothing much at all. It doesn't have much to do with your citizenship though. In terms of your car, I"d figure out which country you want to make your primary residence and register and insure it there. Keep in mind that each state and province makes its own laws on that stuff though and you can find yourself in a situation in which you are legally required to register your car and have insurance in more than one jurisdiction at the same time depending on your residential intentions and/or the time your are spending in the various jurisdictions. Better read up on the applicable provincial and/or state law. The Multilateral Convention on Road Traffic--to which both the US and Canada are signatories--may protect you against double registration requirements (or it may not depending on your circumstances), but don't expect local law enforcement to know anything about it.
Moreover, be aware that the US and Canada are still promising to start sharing entry/exit data on their citizens sometime in the near future, so CBP and CBSA will know exactly how long you are spending in each country--meaning they may decide your residency status for customs purposes differently than you self-declare, potentially leading to an import/export nightmare with your car each time you cross--but that is still a ways off probably.