The uniforms of the time were, like all military uniforms, inspired by the most famous (or infamous) military units of the day. In this case, the period you are speaking of tended to follow a trend that patterned itself after uniforms worn by the French in 1859, as well as the Prussians of 1866 and 1871, and the Hungarian Rebels of 1848-1849. (The so-called "Hussar Braid" that turns up on many uniforms of the era is Hungarian, the Kepi French and Hungarian, the Belgic Shako a French/British design, etc.) This is the same reason that virtually every minor German state had one variation or another of the Pickelhaube (with the notable exception of Bavaria, which preferred a Raupenhelm, and of Hanover, which used a Belgic style Shako, in mimicry of the French and British.)
In short, these are simply what were "trendy" at the time. Uniform fashions come and go, just as they have for centuries.
For an extreme version of this, have a look at the various units formed during the American Civil War, particularly in the early war period from about 1861-1862. You have Zouaves, inspired by France; Kepis, short coats, and Hussar braids inspired by the Hungarians; dress uniforms inspired by the British and French; and plenty of state units drawing all kinds of interesting ideas from European powers. (One unit wore Tyrolean style hats, another preferred Turbans; there was a unit in Mississippi issued with red coats and uniforms in mimicry of the British; another in Kentucky was dressed not unlike Russian Grenadier Guards.)
How did they tell each other apart? Usually flags helped. So too did language. There were problems, however. Due to the extreme similarity of uniforms in 1866, various belligerent powers issued colored armbands to denote the allegiance of their soldiers to either the Prussian or Federal side of the conflict. This seems to have helped somewhat. (It's certainly difficult enough to tell the difference between a Hessian and a Prussian via uniform plates alone - I imagine it was similarly confusing at a distance, especially when one speaks the same language.)
-Doc