Wrong.
I will give an example.
When a foreign airline leases a US-registered airplane, for use in their scheduled services, the FAA (normally) issues, to the pilots that will fly these US-registered airplanes, a US license (no testing required), and these FAA licenses are restricted to only these specific airplanes, by registration number.
Been this way for well over twenty years.
Fact.
Correct! Some of my friends at MXA who are awaiting their uncertain fate have FAA and JAA licenses (or better validations) for N- and F-OXXX registered planes. The only thing I don't know is whether the FAA calls it a license or a validation, but I'll leave that detail for to the bureaucratically gifted...
Usually leased airplanes will keep the registration of the lessor's country, so that it is easier to repo the plane in case the airline defaults on its payments. MXA's F100s were PH- registered for a long time, before they were finally purchased.
There is also an airline in the US (Ryan International I believe) that flies EI- registered (Ireland) 737s. Not sure if they still do, and if those were leased from the (in)famous Irish carrier Ryanair (FR).