Fuji, the CofA registration of an aircraft is a completely separate issue from whether it is being operated for public transport. Public Transport can only be carried out by an organisation holding an air operator's certificate (AOC) using aircraft registered for Transport (Passenger) or Transport (Freight) and flown by holders of a CPL or ATPL. Look closely at the CofA and you'll not find the word 'public' anywhere.
Flying schools are required to have their aircraft registered for Tansport (Passenger) and some private owners who bought from a school may still have CofAs in this category. The difference to a private owner of Private Cat and Transport (Pax) is all down to record keeping and maintenance.