Ownership is a term that is loaded and the argument between socialism and capitalism is actually not really helpful.

For instance, if we removed the concept of capital and implemented socialism, there would still be a concept of "responsibility for". So if there were community bicycles, individuals would still have responsibility if they threw the bicycles in a lake. The community officially "owned" the bicycles, and the individual acted against the community's best interests.

Capitalism as written about by Adam Smith seeks to solve the same problem by putting the "ownership" in the hands of the individual, so that the individual is given incentive to care and maintain the bike.

These are both potentially-effective systems that seek to solve the problem of "who's responsible for this bicycle". There are trade-offs for each.

When we simplify complex market-based problems into choices between either capitalism or socialism, we have created nice ways to describe the problem, but neither of them serves as an effective solution to the problem.

For the solution, we have to dig deeper and that means not being beholden to ideologies that can prevent us from seeing good solutions.