The tax relates to the vehicle and is paid by or on behalf of "the keeper". The seller can leave any tax in place and either come to an agreement with the buyer regarding recompense or else include it in the agreed sale price (if the seller paid the tax and has a call on its value, see below).
Although the car is being sold and it is a stated legal requirement to let the DVLA know the car has been sold, in a peculiarly British way the DVLA nevertheless (and as a contradiction) refuses to have anything to do with confirming/recording ownership (which "has nothing to do with the registration of the vehicle") and there is no official record kept of who owns any vehicle in the UK (apart from individual, private contracts of sale).
The responsibility regarding tax, fines, etc. is what is being transferred, supposedly irrespective of who owns the vehicle - the authorities want to know who is the keeper, not who is the owner (although they may be one and the same, just don't expect them to want to know who is the owner) although they insist informing them of a sale is a requirement. This must be among the handiest/best examples of nonsense available.