Actually, not much. Communism's basic premise is "From each according to his ability to each according to his needs."
Insurance requires everyone to pay the same, but everyone receives benefits according to their needs. Like communism, insurance ends up supporting a large bureaucracy. Neither the communist bureaucracy, nor the insurance bureaucracy is beloved by the populations they serve. A lot of effort gets spent on various forms of social conflict that would have been spent on delivering services in a society where insurance did not exist.
Both communism and insurance promote a zero-sum mentality. Success is defined as getting a larger share of a fixed revenue stream. Contrast this to the capitalist mind set where success is defined by creating value and selling it.
One insurance company tries to impress potential customers by putting showing a picture of it's huge headquarters building. If that insurance company didn't exist, that building might be a hospital full of doctors, or maybe a bunch of houses for people who have been burned out.
If you buy a license to experience a disaster, are you more likely to experience one? Will you feel cheated if you don't?