made their way through the ranks as cavalry officers, and many others were horsemen. Like a lot of military uniforms, there is a lot of tradition attached to them.  The US Army had a command for horse cavalry until almost the middle of WWII.  Even today, in mechanized cavalry units officers and NCOs can wear the Stetsons which hearken back to the horse cavalry days.

And the Wehrmacht and the Red Army actually used millions of horses during the war. For the Germans, it was super important because it saved on petroleum to use horses for logistics.