You could do that in Morrowind too, but it was removed in Oblivion, probably for balance reasons, unarmored characters were already at a severe disadvantage due to lower defense, added to that, in Morrowind they inevitably had fewer enchantment slots than fully armored characters, in short, the game pretty much screwed them order even if you ignore the fact that the unarmored skill didn't even work in Morrowind if you didn't wear at least one piece of armor. Of course, the change didn't really fix the issue, due to some decisions such as removing the unarmored skill, meaning unarmored characters had no way to have any level of defense without relying on spells or enchantments, which armored characters could do just as well, and the absolutely baffling decision to not put unarmored gloves in trhe game, meaning that there was a lack of anything to wear in the hand slot if you wanted to go unarmored. But that's another matter, it's not that you can't make unarmored an alternative to wearing armor actually worth considering for players who aren't interested in intentionally handy-capping their characters, it's just that Bethesda didn't really succeed in doing that in Oblivion, although they did make the first step that would be necessary to accomplish that.
Well, there were those wrist irons you started the game with. I'm pretty sure those qualified as Unarmored, and were enchantable.
Anyway, I never even enchanted my clothing in Morrowind, because of that goofy "Plastic-Wrap" effect. An easy, but probably angering solution to satiate balance+variety, is not allow Armor to be enchanted, but still allow clothing to be worn as underlay. I mean, technically Armor already provides a function, additional apparel's draw, could be the ability to take enchantments.
That's not to say No-enchanted armor would exist, just keeping it reserved for powerful relics.
What about clothing over armor?...
I don't see the options being realistically implementable together, outside of a "This is always over, this is always under" type thing, as in Morrowind.