Does a council tax bailiff know my car registration?

No

Bailiffs enforcing unpaid council tax act under an enforcement power called a liability order.^[1]

The information on a liability order does not contain details of vehicles or goods owned by the debtor.

When bailiffs collect an unpaid traffic contravention debt, or an unpaid court fine for a motoring offence, the warrant of control will give the bailiff the details of the vehicle involved in the contravention.

The contravention vehicle might not be owned by the debtor, for example it is a company car, leased or on hire-purchase, and these cannot be taken into control by bailiffs to pay a debt.^[2]

^{[1] Regulation 34 of the Taking Control of Goods Regulations 2013
[2] Paragraph 10 of Schedule 12 of the Tribunals, Courts and Enforcement Act 2007}