As Woofgang says, anyone can set up a school or college. It's only those providing education, for more than 18 hours per week, to pupils of compulsory school age that have to be registered with Ofsted.
So, for example, those schools of drama or music which provide after-school education for children aren't required to register. (The only exception would be if children under 8 years of age were to be included; that might trigger a registration requirement under the 'childcare' rules).
Similarly, many single-faith schools deliberately choose to provide education for just under 18 hour per week, in order to avoid the need for registration. (Their pupils are officially seen as being home-schooled, with parents choosing to use the services of such schools in order to assist with their education).
Colleges within the (formal) Further Education sector do fall within the remit of Ofsted but others don't.