In many countries you renew your "road tax" or vehicle registration annually which includes third party insurance. Sometimes the number plate itself has a marker, changed annually, to show this. This means insurance is no big deal, everyone who drives a vehicle has it. Different risks, age, whatever, I don't know how it works, but it apparently does. If you want more cover, I presume you can buy it.
What went wrong in the UK? So many uninsured drivers, so many young or inexperienced drivers struggling to afford their premiums, so much complication and hassle at renewal time where you often have to change insurers to get a decent deal. Everything's an extra, everything's a massive risk and they're doing you a huge favour by insuring you at all. Don't dare to be a non standard human or drive something out of the ordinary. All the time being bombarded with TV and other advertising about something which most people are supremely uninterested in (until you have to make a claim) and only reluctantly buy once a year because you have to. The insurance industry takes up, too much brain room in peoples' lives which could be usefully directed somewhere else.
So while you can include bike insurance on your household contents policy, or get it included in membership of Cycling UK or other organisations why would you want to even go near a motor insurance company with their endless lists of exclusions and add ons. Who has a standard bike? Most people modify things to suit themselves so it's a completely different scenario. Plus, of course, insurance is not a legal requirement for cycling.