Dave DuPlantis cited the Saints Row Wiki, but still gave the wrong answer despite the fact that the wiki contains the correct answer, as well as a full explanation.
As it says on http://sr.wikia.com/Garage - the limit is 64 self-added vehicles, in all 3 games. Unlocked reward vehicles are not counted towards that limit. So the limit is 64+X, where X is total number of unlockable vehicles in the game. But you can only add 64 vehicles yourself. If your garage is full with 64 vehicles, unlockable vehicles will still be added.
The in-game help page for Saints Row and Saints Row 2 says the capacity of the garage was 160, but can be proven incorrect by anyone willing to steal 65 cars off the street and add them to the garage.
In all 3 games, there is only one "master garage list" for all types of vehicles, if you add 64 helicopters, you can add no cars.
The previous answer when on to discuss the number of vehicles in each game, and questioned the accuracy of the list on the wiki: http://saintsrow.wikia.com/wiki/Vehicles_in_Saints_Row:_The_Third
That page states that there are 85 vehicles in the game, which is correct, although the list on that page has several more due to duplicate vehicles used in cutscenes which cannot be obtained.
There are an additional 17 DLC vehicles, which are not counted in the 85.
The previous answer stated that "Sports09" redirects to the Torch page - this is because that is what the vehicle is, and there is no reason to have a duplicate article for it. The Sports09 is not counted in the count of 85. Neither are Car 57, Cargo2_m9, Cargo_heist, Eagle_MAD, or HeliMS05.
The previous answer stated "I don't know. Anyway, it's possible that something about my particular assortment of cars limits what I can add, but in the absence of anything official or better documented, I would go with 89."
The previous reply was correct, in that the submitted did not know what they were talking about.
The "assortment of car" does not limit what can be added. The limit is 64 no matter whether it is 64 cop cars, or 64 different vehicles. There is no need for "anything official or better", and ANYONE can spent 20 minutes adding cars to the garage to check the limit. Using cheats will probably speed that up - they also count towards the limit.