The longer answer to this can be found in Section B of the MOT Guide, but in summary: Any authorisation by DVSA allows only the MOT Station legal entity authorised to provide the testing service. If a company is reconstituted in a way that leads to a new company registration and number being issued, then it will be regarded as a new entity and a new authorisation is needed. If, in a partnership, a partner leaves or joins, the partnership becomes a new entity, so again, a new authorisation is needed.
This is also the case if a sole trader takes on a partner or forms a company (see also Transfer of records following cessation or disciplinary action).