Pretty much as the title says.
Different armies did, indeed, use different types of horses. Moreover, within each army, particular types of unit had strong preferences when it came to the breed of horses that they used.
Where entire armies were concerned, the key considerations were climate and diet. Thus, an army located in a cold country with long winters needed horses that were different from those of armies located in temperate or warm climates. Likewise, a country that produced a great deal of grain would use horses that were different from countries that produced less grain.
Within armies, men of heavy cavalry units (such as the French cuirassiers and the German Schwere Reiter) rode on large horses that were good at moving very quickly for short periods of time. This was due to the definitive task of heavy cavalry, which was to conduct cavalry charges in battle. Men of light cavalry units (such as hussars) rode on small horses with good endurance. They needed such horses because their definitive tasks were long-range reconnaissance and raids.
Units of mountain artillery liked small, hardy quadrupeds, such as ponies, mules, and even donkeys. Units of field artillery liked horses of medium size. Units of heavy artillery liked large horses with good endurance. (To imagine such horses, imagine the Clydesdales that pull the Budweiser wagon.)
As far as reserving breeds for government use, many armies maintained there own stud farms, with a view to increasing the supply of the sorts of horses that they liked to use. However, as it was expense to maintain a horse, armies also used various schemes by which military horses could be used in the civilian economy. In some cases, this was a simple matter of providing a subsidy to people who maintained horses of a certain kind and agreed to provide those horses to the army in the case of mobilization. In other cases, military horses were leased to civilians for particular periods of time.
Given the reliance of most armies on these "get civilians to feed army horses" schemes, it was very difficult for armies to restrict the use of particular breeds to military use. With care, they might have been able to preserve particular bloodlines, and restrict the export of particularly valuable breeding animals, but they would not have been able to prevent the export of all members of a particular breed.
If you want to get a quick introduction to the world of military horses, I recommend a book written by my friend Rich DiNardo some years ago. You may also enjoy going to Google Books and browsing through old issues of The Cavalry Journal" (UK) or the Journal of the US Cavalry Association.
As pointed out yes they did and if you are from Europe in most Western Armies they retain Cavalry Breeding Stations which are the evolution of the original Military Studs that bred heavy and light horses to the armies. I am more familiar with Spain and France where both have their Army Breeding Stations, "Yeguada Militar" in Spain and Haras in France. The catalogue of stallions include Andalusian horses (formerly for cavalry units), Arabs and Andalusian-Arabs as well as draft horses (artillery and baggage train). Arabs were for officers and cavalry although they also retained a key role for bred improvement, or to introduce new blood in breeding lines to avoid too much consanguinity, this is still done today. France followed a similar approach although French and British had a preference for Thoroughbreds to improve their local horses. Further North and in Eastern Europe Trekkeners were highly appreciated.
This database lets you track the pedigree of many sires and mares, although has a use in sports and modern horse breeding many breeding lines of modern sport horses come from those State and Army Studs created few centuries ago.