A warrant for an unpaid parking fine has the car's number plate, can bailiffs remove a different car?

Yes

Bailiffs may take control of goods that belong to the debtor.^[1]

Bailiffs recovering a warrant of control for an unpaid traffic contravention debt, or an unpaid magistrates’ court fine, will know the registration of the vehicle involved in the contravention or traffic offence, because it is on the warrant of control.

Before calling at the address, bailiffs will look around the neighbourhood for the contravention vehicle, and will most likely clamp it before knocking the door.

If the contravention vehicle cannot be found, bailiffs can take control of any vehicle owned by the debtor, but they have no ability to search the DVLA for vehicles that have the debtor names as the registered keeper.

Vehicles on hire purchase cannot be taken into control by bailiffs,[2] and if bailiffs take control of a vehicle that does not belong to the debtor, the owner can make a third-party claim.^[3]

Vehicle owners can take steps to protect their vehicles from bailiffs.^[4]

^{[1] Paragraph 10 of Schedule 12 of the Tribunals, Courts and Enforcement Act 2007
[2] See Bailiffs and clamping hire-purchase vehicles
[3] See Bailiffs taking or clamping other person's vehicle, making a third-party claim
[4] See Protecting your car from bailiffs}