It's a custom, proprietary UI library built on Skia. It would be really cool if they released it as open-source, but it's also their competitive advantage.
Just to clarify, we use very little of Skia these days - we draw some basic primitives with it (i.e. rounded rectangles, pngs drawn to a rect) in software rendering mode only. With the new GPU renderer we don't use Skia at all.
Huh! That's really cool. I'm glad that you invested in the UI toolkit---that distinguishes Sublime products from other text editors and Git GUIs by a wide margin.
The new OpenGL rendering path actually doesn't touch Skia at all. It's pretty much custom all the way down to the system APIs.