In this case, there is an Act of Parliament (Section 172 of the Road Traffic Act 1988, as amended by Section 21 of the Road Traffic Act 1991) that makes it an offence to withhold this information. That is the end of the story: Parliament has said it, therefore it is so.
(Note that in the U.K. there is currently a way out: the ECHR. I don't think they cover the right to not self-incriminate, but if they do that would be a possible challenge.)
Remember: in the U.K., Parliament governs by wielding the power of the monarch. That power is formally unbounded and absolute.