A UK orientated answer but I think it's generally the same elsewhere...
ATC do not clear aircraft for a Cat I or Cat II approach. The pilot is made aware of the category of approach that the ground equipment can support and the aircraft will be cleared to use it in whatever way the Ops Manual says.
In practical terms, if the ILS can support Cat II or Cat III approaches, a pilot has assurance that it will be suitably protected for use in such conditions when LVPs are in force.