What kind of SIN is the fake SIN?

It depends what you want and how much you're willing to spend.

The description of the information inside a SIN is as follows:

A SIN is issued by a country or extraterritorial corporation (AA or AAA rating) at the time a person becomes a citizen. This is generally when a person is born "legally" in that country somewhere -- a hospital, clinic, or maybe even at home with the assistance of a Renraku DulaDrone™️. A new SIN will be issued in the case of a change in citizenship. The SIN itself is a string of characters generated using some of the person's vital statistics fed into an incredibly complicated mathematical algorithm. This guarantees that no two SINs will ever be the same. However, this means that anyone with access to the proper software, like law enforcement agents and other government officials, can know several things just by looking at the SIN: your name, birth date, place of birth, and nation that issued the SIN. A set of biometric data including DNA, retinal scan, and fingerprints will also be taken and logged into the system, associated with the newly created SIN. All of this information is then registered with two master databases: one maintained by the country that issued the SIN, and the Global SIN Registry (GSINR) controlled by the Corporate Court. These datastores are among the most secure on the planet, with multiple off-line backups for each.

-- "Gamemaster Advice: Identification", Shadowrun 5th Edition Core Rulebook p. 367, emphasis mine

Whatever the fake SIN is, it's going to be queryable for some kind of information on the place of birth and issuing nation or megacorp of the person it's supposed to correspond to, in addition to whatever other supporting information there is.

But what is the fake SIN going to say when it's queried, exactly? That all depends on its rating. You can look at pages 367 and 368 for what kind of information a fake SIN will have, depending on its rating, and what sort of information various rating scanners check against. For example, the simplest fake SIN, a rating 1, has a valid checksum but all of its supporting information is patent nonsense that any serious scan will likely flag - its issuing nation may well be TEST_NATION_DO_NOT_USE, or somewhere on the other side of the planet.

Rating 2 (close nationality) and rating 3 and above (nationality match) should be understood in relation to whatever you wanted to claim to be when you bought that SIN - if you wanted to be identified as a UCAS citizen or a Renraku drone or whatever. Rating 3 would match your claim, and rating 2 would be some other valid nationality who could have moved to the UCAS or been employed by Renraku.

A sufficiently sophisticated fake SIN (all rating 2s, some rating 1s) will be registered in its purported place of origin and in GSINR, and given that there's no difference in availability or price between corporate and national SINs -- that they're not even broken out individually on p.443, just treated as the same "fake SINs" -- it can probably be assumed that injecting it into GSINR is the tougher part of the fake.

If you had an unlimited budget and grabbed the cream of the crop, a rating 6 fake SIN, it would contain biometrics and vital information matching yours and identify you as a legitimate citizen of whatever issuing organization you wanted - corporation or country. Lower-rated fakes make various compromises with this ideal, though generally it isn't until rating 2 that you might not be able to get the issuing organization you wanted.

Fake SINs belong to the organisation you designate them to

A fake SIN carries the required markers to pretend being real, and works because checking the balkanized records of all the megacons and governments is a pain and costs time. So the checking software usually only does a basic scan on the surface. Depending on the rating, there is more than the basic functionality of "something claiming to be a SIN is present", and even might have fitting entries in the relevant SIN database. That means a higher rating withstands a more thorough check.

Now, assigning which 'citizenship' and other data the sin claims, is all fluff and you need to discuss those specifics with the GM. While technically only nation states and corporations that are registring with GSINR are allowed to give out valid SINs mutually recognizted in all jurisdictions, not all real SINs are actually registered or accepted worldwide. The regulation on SINs stems from the 2042 Business Recognition Accords, which granted Megacons quasi-government status, but was updated after each crash.

Similarly, while some SINs fit better to certain spots than others, there is rarely an effect on the game. The only difference is the validity or accessibility of some SINs in certain areas, especially for some sorts of people. For example, a Centaur's SIN from the Pueblo Corporate Council is all valid there, but if he goes to the CAS, where Centaurs are legally animals, the SIN is worth nothing.

Besides the governmental issued national SINs and Megacon issued SINs, there are also some other special SINs with limited validity. Berlin has a very specific type of SIN that does not make you an ADL citizen, the VolksSIN. It's worthless outside of Berlin.