When Part-FCL comes into force (17th Sept in the UK) the requirement for approval of aircraft disappears. In any case, the idea that individual airframes need to be approved is more about jobs for the CAA boys than compliance with JAR-FCL.
AFAIK, the UK is the only JAA state that interprets the requirements in this way; normally the
type is approved for training and that's it.
Any FTO (forget ATOs for the foreseeable future) with approval for the CPL and IR can, quite easily, get another aircraft approved but, if it is of a type that they don't normally use, they will, in the UK at least, have to develop (or, more usually, copy from another FTO) class rating theoretical knowledge examinations related to the type for approval by the Authority. Consequently, if the FTO do not already use the type, it is not worth their while to go through the approval process for a single trainee.
Normally, a training aircraft would need to be maintained to public transport standards but, as in this case, if no payment is being made for hire of the aircraft, this becomes irrelevant.