In fact, most people who live in a house that is insured probably are covered by personal liability insurance, which includes damage caused whilst cycling. Such insurance is extremely cheap because cycling is an extremely low-risk activity. For example, the personal liability part of my house insurance covers anyone in the house up to £2 million, and I get £10 million of cover thrown in free with my CyclingUK membership.
People who don’t have such insurance would need to be pursued through the courts just like anyone else who causes damage – whilst carrying a ladder, for example. Motor vehicles, on the other hand, are required to be insured because they bring such a massive risk to the public: