You do ask a lot of questions NewbieFlyer. Wondering whether you've read the archive threads?
To answer your question it depends on how it is done, and it is also done in other industries where separate dub divisions market/sell the same or a variation of a product.
Sometimes schools do it because they know that some of the Europeans don't trust Americans so have a European Subsidiary which basically sells a European or even an FAA course but you will find that they are effectively selling the "umbrella" companies services ie it will be the umbrella companies aircraft and instructors.
Basically sub contracting work to a company with the relevant approvals.
You see this a lot in countries such as Italy and Scandanavia where students pay a company in Ireland, Italy and Scandinavia vast amounts of money, and then they subcontract the flying to a US school for less than half the cost.
PTC/FIT, Pelican and the school in FLL which went bust, Italians/OFT, EFT/Aviator....the list goes on.
Technically the student when in the USA is a student of the US school, with them providing the flight training, with another company managing it, and may even have administrative staff looking after their students interests.
Same occurs in Military training with civilian contractors.
Hope this clears it up.