I belleve a green card holder is for the purposes of aircraft ownership the same as a US citizen or US trust, and not the same as a foreign national.
(However on a more basic non-aircraft related level, the green card holder must show good faith in being a US resident to maintain the green card, for example no absences from the US of more than a year)
Yes, green card holders can be on registry. They’re considered legal aliens.
US Registration Of Non-US Owned Aircraft : an excellent explication of the different options can be read here :
[ local copy ]
Honestly, I don’t “get” all the anti-foreign reg. BS going on around GA.
The boating community don’t seem to be bothered one iotta about the practice of “foreign” registration.
Just take a stroll down any marina gangway in the Med and you will see plenty of registrations from all over the world !
Just take a stroll to any commercial harbour and you’ll see cargo boats flying non EU flags, (but never US), engaged in trade between EU ports.
The whole N-reg topic is unfortunately wrapped up in complicated politics and prejudices.
Among European [self proclaimed] intellectuals there is a ritual dislike of the US which gets accused of polluting the old European languages with English, and infecting their cultures with MacDonalds, boilerplate CGI movies, and such like. In aviation specifically, if you go to some EASA etc conference and/or watch the various videos of the proceedings, you see this spill out with statements like “we are Europeans, not Americans, and we need European regulation, not American regulation” which is mostly banal nonsense since most European aviation regs are originally the American ones but edited to protect the different landscape of commercial and political interests over here.
The N-reg scene has been highly provocative for many years because so many pilots (myself included) did the IR via the FAA route, to avoid the rigid convoluted path over here, thereby depriving European FTOs of business and in turn depriving the European CAAs of licensing income. This angle is declining now, not because the CB IR is significantly easier to do ab-initio but because the FAA route is harder, with written exams and generally checkrides too having to be done in the US too.
And being N-reg enables the use of FAA STCs which deprives European Part 21 companies of income for EASA STC development and sales, and (like ex RAF people used to go to the CAA) EASA is full of former Part 21 company people 
I have had some comms with them years ago and the attitudes were pretty unbelievable. The EASA STC situation is much better now if you stick to bog standard boxes (via Garmin having basically cornered the market ) but it is easy to see the prejudices.
And finally there is stuff like N-reg owners being accused of tax evasion. My old friend Stefan (sadly no longer with us) used to tell me he moved his newly acquired TB20 from the (priceless) M-reg to D-reg for exactly this reason.
So the prejudices just keep piling up, and those who don’t know the background accept them readily.
In decades to come, N-reg will probably die out, as the existing community gets old and retires. I don’t come across any significant number of people doing it today, and one serial N-reg SR22 owner told me recently how he will never do it again (he does have both sets of papers) because his last plane took many months to sell and he had to more or less pay for the transfer to G-reg.
I don’t think shipping has this problem because (a) flags of convenience go back for ever and (b) America is not a player 
Peter wrote:
And finally there is stuff like N-reg owners being accused of tax evasion. My old friend Stefan (sadly no longer with us) used to tell me he moved his newly acquired TB20 from the (priceless) M-reg to D-reg for exactly this reason.
It seems to me the M-reg for aircraft of MTOW below 2730 kg (which a TB20 is) is only for residents of the Isle of Man; according to https://www.iomaircraftregistry.com/services/register-an-aircraft/ even a >=2730 kg but <5700 kg aircraft is “may be considered if sigificantly economically good for the IoM”, not “may be registered”. Or did you mean Stefan moved from N-reg?
The IOM indeed implements the situation as you list, except that
Stefan (sorry I didn’t not make this clear) would refer to M or N as having the same problem.
