I always think of light colored beers having a crispness to them with a light finish. Oatmeal gives stouts and porter a creaminess and perceived thickness to the mouth feel that contributes to the heartiness of the styles. That thickness in a lager would be off putting.
It works great in places that you want to add some heft, but not the sweetness that goes comes along with crystal/cara malts. I have a ~4.5% pale ale with 10% oats that is pretty tasty. It does add a bit of haze (which you dont normally see in a stout). It is also a pretty common ingredient in Belgian Ales like saison and wit.
I could see it working well in a bigger/richer lager, but I havent tried it.