I am interested to know the answer to this as well. But there is something else I’d like to know…
In market economies where corporations are a construct of the law, insurance is often incentive for companies to cover the cost of externalities, like consumer safety, worker safety, and environmental protection. A corporation in a competitive market will do cost/benefit analysis, if they are polluting a river and insurance against litigation increases, they may find that it is less expensive to pollute the river than it is to cover the cost of litigation and take measures to reduce pollution. In practice, insurance in market economies does very little to actually keep people safe or to prevent pollution, but it does have that effect sometimes.
But I would guess that in the USSR, insurance was only a form of risk pooling and could not have incentivized industries to cover the cost of their externalities. Maybe they thought that the people in the soviets directly effected by pollution would have the right to vote for laws that would have forced corporations to not pollute. But based on what I know of the USSR, I would guess it probably never worked out that way in practice, it was probably not much more effective than market economies with insurance companies.
There were insurance requirements according to this archived
https://www.nytimes.com/1989/08/27/travel/behind-the-wheel-in-the-soviet-union.html. It doesn’t really go too in depth about how the insurance works, just that it is a requirement. I am pretty sure it was state owned because pretty much everything was, but I couldn’t find a super credible source on it beyond people online just saying that it was state owned.
It was extremely hard to get permission to own a vehicle, and it was even more difficult to get gas and parts to maintain the vehicle. Usually only Soviet officials and well connected people had cars. It was such an issue that normal citizens couldn’t get cars there was a comedic satire film called Beware of the Car where an insurance agent would steal cars from politicians and scammers and sell the cars to give donations to orphanages.