Possible for some makers and not for others.
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There are plenty of other characteristics by which to identify the manufacturer of an SKS rifle.
The easiest way to identify the model is to identify the country of origin. The simplest way of doing this is to look for the importer's mark. In some cases, it may be on the side of the receiver. In others, it'll be on the bottom of the barrel, and you'll need to unfold the bayonet and possibly remove the cleaning rod to see it. If you do not have any such markings on your rifle, it may possibly be a war trophy which had been brought into the country by a soldier returning home from Korea, Vietnam, etc, and could be worth some serious money.
Here's some more identifying features to help you:
- Soviet models have two distinct arsenal markings - Izhevsk or Tula. Izhevsk rifles will have a marking with an arrow inside of a triangle, along with year of manufacture, on top of the receiver cover. Tula rifles are marked the same way, except their arsenal mark is an arrow inside of a star. Many of the Russian rifles imported into the US actually came from Poland, and were refurbished with a laminate stock, and, in some cases, a replacement bolt (these will usually be blued) replaced the original, and the numbers were force matched to the receiver.
- Chinese rifles will have Chinese lettering on the side of the receiver, in front of the serial number. They're also commonly known for having a spike bayonet, although earlier rifles did use a sword bayonet. There are MANY variations of the Chinese SKS, which may feature thumbhole stocks, different barrel lengths, and some were manufactured to accept detachable AK magazines.
- Rifles from the former Yugoslavia tend to be built a bit more robust, with thicker receiver walls. They also typically lack chrome lined barrels, although some of the very latest production rifles did have this feature. The M59 was more or less a direct copy of the Russian SKS, while the M59/66 featured an attached grenade launcher (which was actually NATO spec, and not Warsaw Pact spec), a gas cutoff valve, a grenade launcher sight above the gas tube which flips up, and also flip-up tritium night sights. The only markings from Zastava (the manufacturer) are the serial number.
- Albanian rifles are very distinctly different from other types of SKS. They have a different magazine design, they have an AK-styled charging handle on the bolt carrier, and they have a spike bayonet similar to the Chinese rifles, as well as furniture uncommon to any other manufacturer of the SKS, which extends all the way to the gas block. There is also a sling swivel, positioned to the side, on the buttstock.
- North Korean rifles are extremely rare, and were known as the Type 63. On the top of the receiver cover, they will be denoted by a star inside of a circle. Underneath this, the number '63', followed by a Hangul character.
- East German rifles are also extremely rare. They accept a sling which actually goes through the buttstock, similar to the Mauser sling.
- Romanian SKS rifles will have the arsenal marks on the side of the receiver, underneath the opening for the bolt carrier.
Vietnamese SKS rifles are also extremely rare, and you would need to get in touch with someone who knows more about them to identify if yours is one.
If you bought it from a gun store, it's most likely from either China or the former Yugoslavia, with a still somewhat fair chance that it could be Soviet, a reduced chance of it being Romanian or Albanian, and a very low chance of it being North Korean, Vietnamese, or East German, unless you paid enough for it to buy three or four SKS rifles of any other nationality.