US N-number a/c abroad

While I was watching the news from an international satellite, they showed a
single-engine Piper that made an emmergency crash-landing on a highyay in
Greece. When I saw it's registration number N-40899 I thought that it
actually came from the US. (That's a heck of a trans-atlantic trip for
single engine!, - no kidding the old aircraft failed, I said)

I then searched the A/C registration database on landings.com, entered its
tail number: N-40899 and I found that the A/C is registred in the states,
BUT it's owners and operations are based in Greece! How can that be? Does
this mean that this aircraft is under FAA rules and jurisdictions? What are
the advantages/disadvantages for a foreign owner to do this?

Would the opposite be true? To register an aircraft in Greece, permenantly
import it to the states and have it flying with an SX-123456 greek tail
number, for example?

I know that this is common practice for US ships to sail under a Bahamas
flag, but I didn't know that it's also true for light single-engine general
aviation aircraft!