Did the Terminator who Reese even was?

The movie itself does not allow a definitive answer, but my guess is that The Terminator would have immediately known to 99.99% certainty that Reese must have been sent back in time, like itself. Only such a person would've been guarding her continuously and have possessed the requisite foreknowledge to respond so quickly to its first motion to threaten her. The only other possibility was that Reese was an undercover cop assigned to catch the serial "Sarah Connor killer," but observing Reese using a sawed-off shotgun would have eliminated that possibility for The Terminator.

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Being that the machine was programmed to terminate Sarah at all costs, my thought is, the Terminator said to itself, "One of the puny, insignificant humans is interfering with my mission. Must get rid of interference. Then terminate Sarah Connor as programmed."

I don't think so because if that were the case, the primary mission might've been to first kill Reese, & once he's gone, terminate Sarah.
As it is, Kyle's just a guy helping her run from him.

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The movie doesn't give us an answer, but I would guess that it did not because that seems to be outside of its purpose. Its primary (and only, I guess) purpose was to destroy its target and it would only be concerned with Reese as much as he posed a threat to completing that goal, but beyond that it probably didn't care. I mean, what difference would it make to the Terminator if it could determine whether Reese was part of the resistance or not as long as he was limited to the same weapons everyone else was? I mean, unless Reese had access to some vital information that severely threatened the Terminator's mission, it seems to have little reason to try to figure out who he was. This is all a guess, of course, all based off of what evidence we see in the movie.

If you include what is seen in Terminator 2, however, then it seems not only possible, but also likely that it did (but it also seems that some people do not accept things in the later movies as "canon", so this one is up to you). The Terminator in this movie constantly demonstrates a great degree of situational awareness and threat assessment, so it was capable of complex analysis. Remember, it was able to determine that John's foster parents were dead from one brief phone call. This would have required it to analyze a lot of bits of information and organize it and string it together to create a conclusion, so assuming that the first Terminator could do the same with Reese isn't unreasonable. But maybe all this stuff was added in the sequel, it might not be valid within the context of the first movie.

Overall, I guess it depends whether you are asking based solely on the first movie or if you are also including the later movies (I saw the first three, but the first two are the only ones that exist in my opinion). We see no evidence to support this in the first movie, but a lack of evidence doesn't necessarily mean something doesn't exist, so it is possible and there is no definitive answer.

The Terminator is a machine designed to destroy. It has a sophisticated A.I. so it can mimic human behaviour as well as possible, to optimize its chances to terminate its target by being undetectable until it's too late.

It is not necessary for its mission, its programming or terminating ability to ponder, think or know where its enemy came from or why it is there. It only 'cares' (and I use the term loosely) about accomplishing its mission and programming, and the only function it uses the A.I. capabilities for, is to aid in this terminating mission. Everything else is irrelevant, otherwise it would sit and ponder every flower's origin it sees.

I am saying it makes absolutely NO DIFFERENCE to the terminator, whether Kyle Reese came from the future, from a spaceship, from underground, from Massachussets or from the Arctic Circle. All it HAS to know is that Sarah Connor is to be terminated, and whatever obstacles are in the way, has to be eliminated or removed in any way possible.

It doesn't NEED to know more detail about that situation, it doesn't necessarily even BENEFIT it or its mission to know that. It's not meant to be a philosophical companion, it is meant to be an efficient killing machine. Why would it ever think, ponder or need to know anything about Kyle Reese's origins, and why would its programming 'care'? Would it try to kill Kyle differently, if it knew?

I don't think it had any knowledge about that - the resistance is probably careful to not reveal their faces to the machines so they can't be mapped and catalogued easily, and Skynet had no reason to install such possibly distracting information to a ruthless killing machine. Would a soldier benefit knowing the whole family history of the victim she is murdering?

Did the soldier that sniped someone from three miles away (or whatever it was - I think it was some kind of a record) know the family history in great detail of the now dead victim of his bullet?

To the T-800 Reese was just a random guy trying to save Sarah. It would not have known, unless skynet knew ahead of time and added the file to the T-800's cpu but it didn't have time since Reese came after the T-800 was sent already.