the following info is taken from the book “Плакат Госстраха”, Moscow, 2012.
All insurance in USSR was done by a monopoly, Госстрах, established in 1921. According to the book, the first time (presumably, since 1920s) private cars in USSR were insured (by Госстрах) in 1946, specifically, 99 cars. The number went up to (approx) 1700 by 1949 and to 188500 by 1969.
This forum post may interest you.
Fascinating. Thanks!