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I have faced 1...c6 as a reply to my 1.d4 quite a few times recently, and this makes me wonder why people play this as oppose to 1...d5.
In addition to playing the Slav with 2.c4, I have an option of playing the Caro-Kann with 2.e4, so I have an extra option and black has nothing in return that I can see.
1...c6 happens at top level, but fairly rarely (9th most popular response, behind 1...Nc6 according to lichess database).
Why do some players choose this move order?
Of course you must be ready to play the Caro-Kann if you answer 1. d4 with 1. ... c6. But that is a sound opening - so why not, if you already play it against 1. e4? You will probably be better prepared for the Caro-Kann than your opponent, who after all started with 1. d4. If I was a d4 player, I surely would play 2. c4, hoping to enter the Slav. In that case, blacks most popular choice is 1. d5, but other moves are also possible. Black might play for a setup with d6, g6, Bg7, with or without Nf6.
There is also the interesting Malinoise defence, 1. d4 c6 2. c4 b5!?, which has been played by GM Rogers from Australia and IM Jadoul from Belgium. This should at least be a nice surprise weapon.
Playing 1. ... c6 might also be an idea against London system players, e.g. 1. d4 c6 2. Nf3 Qc7!? has been played a few times - not the best of course, but at least it is a nasty surprise for any white player who planned 3. Bf4. Might be worth a try against certain opponents (especially among London system players, there are some who don't know how to play without their beloved Bf4) The accelerated London system 1. d4 c6 2. Bf4 could be answered with 2. ... Qb6!?, when white has to decide whether or not to sacrifice the pawn on b2.
You're right that players who choose 1...c6 must be fine with the Caro-Kann, which is one reason why it's not that popular. However, there are some people who are fine with the Caro-Kann, and so the move gets played occasionally. In the case of 2.c4, there aren't many benefits I can see for Black. He has the option of playing a la King's Indian with ...Nf6, ...d6, etc, but it wasn't necessary to start with 1...c6 to do this.
There may be some minor benefits to 1...c6, but they're not obvious to me. This is why the move isn't played often.
I would add one little thing to Inertial Ignorance's answer: The reason it is so unpopular is probably that the Caro seems more passive than the Slav, and I say this as a long-time Caro player. E4 openings, being more open games, give white more chance to pressure black is what I mean. Still why let your opponent dictate what line you will play when you can be the one to choose?
Convenience.
At sub-master level, most people don't have a broad opening repertoire, they will always play one specific defense against 1. e4 and one against 1. d4. Most of them will also try to reduce the amount of theory to learn.
Some will pick both the Caro-Kann and the Slav as these two defenses for some transpositional synergy. Then the idea behind 1...c6 is that if you always play that move anyways, either on move 1 or move 2, you might as well always play it at move 1, since you are fine with either 2. c4 or 2. e4. By not occupying d5, you are actually in a way daring White to play one of these moves - instead of some weird Queen's Pawn system that you are not as comfortable with.
To throw off your opponent.
At GM level, giving away your intentions to play only very specific openings can be seen as close to a mistake - you allow your opponent to freely choose the opening they have prepared better, which on average will probably not help you. But it's a strong psychological message: "I am confident I can beat you anyways", which can work in its own completely different ways. Sometimes, people like Carlsen even open their game with strange things as 1. Na3 and still win - 1...c6 can be seen as something between this and a "normal" opening.
