No
Unless the vehicle is the contravention vehicle in a road traffic debt or involved in getting the driver a magistrate's court fine, there is no other way the bailiff knows whether you own a car.
If the bailiff is enforcing an unpaid traffic contravention debt, then the vehicle registration involved will be on the warrant of control. Before attending, the bailiff will look around the neighbourhood for the contravention vehicle.
If the bailiff is enforcing an unpaid court fine involving the use of a vehicle, the vehicle registration will be on the warrant of control.
Bailiffs cannot search a person's name on DVLA to see whether they are the keeper of any vehicles.
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