I did exactly that in a 4e game I ran, a while ago, to a slightly lesser extent: For in-game reasons, only humans and elves were allowed as playable races, meaning, the game world simply was not populated with gnomes, halflings, etc.
However, I knew perfectly well that some of my players wanted to play certain races for mechanical reasons, in combination with certain classes. So I told them up front that they could take any race they wanted, as along as they "skinned" as human or elf.
Where it would have gotten weird (and therefore was my only not-allowed restriction) would be if someone had chosen a dragonborn, because I would have had to justify a human with a breath weapon. But that was not a mechanical issue, it was, itself, a cosmetic issue.
In practice, it was fine. And by fine, I mean we experienced no negatives at all, precisely because this is a cosmetic issue.
I realize you're asking about 5e and I'm answering in the context 4e, which are different systems. But they're not so different that I would expect any problems out of 5e.
Just do be careful, as the comments and this answer suggest, to make sure you're not accidentally cherry-picking some mechanics from both the "skin" race and the mechanical race. And be careful that you're not setting yourself up for something really weird and difficult to justify in-game, like a gnome with a breath weapon.