Not in many places. My department has enough cars for every officer and a handful of old spares. No point in returning it when it's not needed. You'd think that doing that would be a waste of money, but it's really not. Shared squad cars average about 80 to 100,000 miles before they're just flat out worn out. The assigned cars will make it to 140 or 150,000 miles easily. People take better care of cars they're assigned to. The few exceptions to the rule are easily identified and those officers will forever get the oldest, crappiest cars to ruin.
This! In my department, the officers who live in the city can take their cars home. To take a car home outside of the city limits, you have to be "on-call". I don't live within the city I serve. When I was in patrol, I took my pickup back and forth to work. I'm currently assigned to the traffic division as a crash reconstructionist and "on-call" for critical and fatal accidents. It's nice, but sometimes, when the phone rings at 3am, it makes me wonder if the take-home car is worth it.