I'm fairly sure that as far as MOT and DVLA vehicle tax classes are concerned, a taxi* is the same as a private car. Any licensing as a taxi, and any tests over and above MOT standard is up to the local licensing authority (mostly local councils - although London is of course different, and the Public Carriage Office is now part of TFL rather than the Metropolitan Police like it used to be.)
Some councils' tests can take the place of an MOT test if DfT accept that the test is equivalent to an MOT test or more so.
* - technically, the term 'taxi' has no legal status, a vehicle is either a 'hackney carriage' (that can 'ply for hire' on the streets or cab ranks) or a 'private hire vehicle' (which can only be booked in advance via the operator's base.)