Kind of. The browser WebGL implementation compiles them, producing shaders suitable the native driver shader compiler (conversion from OpenGL to D3D, rewrite texture/buffer access for bounds checking as necessary, etc).

Ah yes, you are right, WebGL shaders are translated first to GLSL or HLSL with a lib like ANGLE (https://github.com/google/angle). And that source is then send on to the drivers which compile it again to assembler for the graphic card. And there again are some differences between GLSL/HLSL it seems. If I remember right in GLSL the drivers do the work while in HLSL some DLL from MS produces another intermediate format (which I just found out seems to be even open source: https://github.com/Microsoft/DirectXShaderCompiler).