It's exactly like that: It's the car that has to be insured. Every person with valid driving license is allowed to drive it and is covered with insurance. Insurance company can decline paying the damage only if the driver has been drunken or in drugs.
For me the UK system sounds very peculiar and tortuous. I mean causing a lot of trouble to car owners and has plenty of minefields.
Yes its typically British. Complicated and poorly enforced.
The system in Finland requires government intervention forcing Insurers to give a price thats an average for every one. It benefits high risk drivers and cars, but lower risk drivers pay more than they might. But its still a good system. They have something similar in Bulgaria . Every registered car must have a minimum insurance sticker. Easily spotted an enforced. Issued by private insurance companies but at a regulated price. If the car is uninsured for ,I think 3 months' the car registration is cancelled and they have to go to the police to get a new registration plate. If you want fully comprehensive insurance you still have to buy the minimum sticker, but buy an extra top up policy,sometimes from a different insurance company. But there were complications. The minimum cover is valid in every EU country. They found thousands of Romanians and Greeks were running cars registered and cheaply insured in Bulgaria on a permenant basis in their own countries with lots of insurance claims.

Reality is some drivers, cars , locations etc are a much higher risk for insurers. They want to take money with the lowest possible risk , or might gamble on a risky driver for a bigger premium.
In the UK all registered motor vehicles are assumed to be in use and must be Insured. The Insurance Industry updates a national database of the vehicle currently insured. If you take the car off the road onto private land its the registered keepers responsibility to regularly inform the authorities it is off road (including public places such as car parks) and need not be insured. (SORN - statutory off road notification) Needless to say this system is widely ignored or abused, and the UK has many uninsured drivers. Some insurance companies offer protection from uninsured drivers , but of course legal drivers are paying extra because of uninsured ones.
The car must be insured but the driver must also be insured to drive the car. Either as a policy holder or named driver on the policy. Some policies allow the policy holder (but not named drivers) to drive someone elses car ,usually at minimum cover. And garages and other companies have special insurance.
Some insurance companies are large financial institution with billions of pounds in assets, able to pay any claims.
But many authorised insurers are small organisations of investors who take on a limited risk in return for premium payments. They may not have offices or many staff. They may work under the Lloyds system of syndicates.
They are groups of wealthy investors who ,in return for a share of the premium absolutely guarantee to pay an agreed in amount,(maybe in the millions ) if things go badly wrong. But they dont actually have to put the money into the syndicates bank. But if necessary the poor mite might need to sell one of their mansions to meet the debt .. Meanwhile they keep their millions invested elsewhere. Double return on the same money

Thus the uk have " Insurance companies" such as Hastings direct who are not actually approved Insurers in the strict sense. But they are regulated by the financial authorities, and might have a small approved insurance companies in their organisation. They have large staff, advertising budgets, administer claims etc,. Much of the premium you pay will be for their administration fees. But ultimately they spread their insurance risk by using a panel of smaller approved insurer syndicates to actually underwrite the main risks. Some syndicates prefer low risk drivers, for a smaller, but safer income. Others are willing to take risks for a bigger profit. But nothing is forcing them to insure a particular car or driver.
Then there are Brokers, such as Adrian Flux. Similar in many ways I think. I'm not sure of the exact distinction but I think they tend to be more independent, and the better ones will seek out a good deal for a customer by going to many insurance companies, not just the few they normally work with. They work on commission.
Then there are 'comparison ' web sites such as 'Go compare, and 'compare the market'. These operate a search engine. You enter details, on their site and this is input into numerous other data bases. They then present a list of offers from various insurance companies, including some offers from other 'Insurers' such as Hastings Direct. When you chose one you are directed to that companies web site and deal with them directly. The comparison site has limited further involvement beyond collecting their commission.
I hope this helps. I may be wrong in some details.