I would not say "most" but many people owned them. In the big cities, especially children (I am not talking about rural areas). Streets were NOT crowded with bikes
(cannot even compare with Denmark or China, and even with Germany). Bikes were affordable and available, but few adult people in the big cities used them for transportation.
They were used more for recreation rather than transportation.
About the reasons one can only speculate. First is the climate: a bike is not a good transportation in winter, and winter is very long in most places of Soviet Union.
There was almost no special bicycle roads, and the risk to be hit by a car was high. I would add that most Soviet cities had very good and very affordable public transportation systems.
When I was a child I lived there (in the 1960th). I had a bike, most of my friends had them too. But very rarely one could see an adult riding a bike in the streets of a city.