This has happened a few times in my D&D games, but not enough to be anywhere near the standard. Orcs, drow, etc are semi-common options, but they are usually mixed in with more conventional elves, humans, and the like. An "all monster" campaign is overall very rare.
Way back in the 3.5 days I ran a campaign where the PCs were various kinds of monstrous natives of Faerûn's Underdark, using the sourcebook of the same name for inspiration.
It was pretty standard fantasy, albeit with a more anti-heroic bent. The PCs were effectively mercenaries from various city-states exploring dangerous caverns, sunken crypts and catacombs, mushroom forests, and strongholds of the worshipers of chthonic deities for wealth and prestige.
It was fun, but didn't last long on account of several factors. As we were using Savage Species, the inter-party balance was broken as hell. If anything, I think that 5e can do this better, as there's a wealth of good resources for relatively balanced monstrous races. The big thing is that a lot of the more powerful and problematic abilities are nerfed, if not outright axed.
These all sound like interesting campaigns. It does remind me of how 3.5 Drow's spell resistance turned out to be quite the pain. By RAW, you needed to spend a standard action to lower it, so so oftentimes our Underdark party would forget to lower it and then realize that they can't buff themselves up with spells. The ideal mode of play would be to lower it before an encounter or ambush, buff, then charge into battle, but that wasn't something the group could prepare for so it just became a pain in the butt.
