Was it just a coincidence? I know the Union and Confederates were drawing from the same military traditions and physically taking from the same armouries so their internal similarities don't suprise me. I also know they barely had standard uniforms as such and they were pretty ad hoc, especially in the south.
But was there some deliberate attempt to be inconspicuous on the battlefield, and understanding that it was just good standard practice? And if so what culture and previous experience informed this break with European traditions at the time? If it wasn't a coincidence was there something unique about the American battlefield that made camouflage expedient?
I know camouflage was pretty much an ancient concept and had seem limited use by everyone in the gunpowder era for sharpshooters etc, but it also seems it was never considered broadly practical and necessary until smokeless cartridges were introduced (and in many cases for a long time after). Except in the United States where the plain uniforms are there in the 1860s, along with all these other 20th century premonitions like widespread use of trenches, steam power, telecommunications etc.
Also, did any European observers remark on the American uniforms at the time?
First things first, both the federal and rebel armies were uniformed by regiment, not by any central authority. There were a huge number of different uniform types, with different colors and cuts and headwear. As the war went on, availability of certain dyes as well as traditional US army uniform trends tended to somewhat standardize the federal army as wearing a dark blue coat of some kind with sky blue trousers, neither of which I'd exactly describe as "drab." Many regiments, especially at the start of the war, even chose extremely flamboyant uniform types like the Zouave style uniform, with bright, baggy pantaloons and heavily embroidered round jackets and fezzes, which were considered flamboyant even in the very flamboyant 1860s.
Apart from the specific color choice, the combination of wool trousers, a wool sack coat (sometimes frock coat), brass buttons, and a kepi was deliberately modeled after the French army, and was considered a practical and inexpensive uniform. Very few uniform decisions were made with the idea of keeping hidden, that's not really how armies operated in the 1860s. I've talked a bit about some of the reasons uniforms trended toward camouflage or drab, earthy colors here.
For the rebels, the simplest reason that they wore drab uniforms was the unavailability of dyed wool, and the scarcity of access to good wool for uniforms in the first place. Given the choice, I'm sure rebel regiments would have advocated for more colorful, more professional-looking uniforms than they ended up having, but for an army that regularly had trouble equipping their men with shoes, let alone coats and waterproof blankets and normal blankets, access to colorful or flamboyant cloth was a lesser concern. Men wore what they could, not what they wanted.
So in short, no, there was seldom any attention given to remaining hidden or camouflaged on battlefields. That's not the way war was fought at the time, and uniform choices had as much to do with tradition, professionalism, and access to good cloth and good dyes as they did with any practical field concerns, of which "having clothing" was often more important than "having uniforms."