Exactly 1199 cars at the moment but the number does fluctuate.
I think I have at least one of every car possible and then some. Don't hold me to this because I haven't checked for about six months. At one point I went through every garage, buying every car (most I didn't even want). I did this for about a week just for the heck of it. I would have had many more but it got to be a hassle moving cars to that stupid stockyard. That stockyard seems to take eons to load and save with slightly over 715 cars.
In my main garage I keep the number just below 500 (due to game limitations), so if I buy a new car or win a new one I usually sell it off instead of saving it. I have to keep selling off cars as it is because I can't have more than 50,000,000 credits and when winning seasonals in first place constantly with 200% multiplier, that figure is always being exceeded. At one point I was racing and getting nothing because I was tired of selling off cars but I was annoyed of that "You can't have more that 50,000,000 credits" popup appearing. Now I usually sell off the 20,000,000 credit Ford Mark IV and the 20,000,000 credit Ford GT for a bit of cash and then buy them right back to fall back to 10,000,000 credits. I have a little more than 2,000,000,000 credits won at this point. Kind of pointless actually. It's just something to do when I get bored.