Can someone else insure your own car and add you to their own quote for the same car?

In general, yes. Parents can insure a vehicle and add their qualified children.

Friends can insure a car and add a qualified buddy.

The premiums may change drastically when the details of an additional driver are included.

If full details are not disclosed, the insurance policy may be void and that opens the door for prosecution for not having insurance for the insured person AND the additional driver.

You cannot add someone to your insurance if they have been banned or if they have a medical condition that prohibits them from driving. Other factors may come into play. The only way to fully know if the additional driver is covered is if the insurance company speak to them directly.

Ownership of the vehicle is separate from insurance. Ownership is also separate from the Registered Keeper.

Mr Smith does not own a car. Mr Brown owns a Saab 93 but he lets Mr Smith use it to commute to work. Mr Smith is made the registered keeper so that traffic offences, parking, speeding, etc. are not attributed to Mr Brown. Mr Brown insures the car but Mr Smith is an additional named driver. Mr Brown retains the title to the vehicle.