Ah, I see you are using GLUT. At least w.r.t. GLUT and GLX Winex should be redundant since GLUT is available for both platforms, just recompile and it should work.
For GLUT the window attributes are defined in the glutInitDisplayMode call. GLUT_DEPTH seems to be the token, but there is no direct control over the depth buffer size. This seems like a weakness of that API.
What you get will depend on the details of the code inside GLUT that creates the visual attributes on each native platform. The alternative is to go with another approach to windowing. Ofcourse I don’t actually know what GLUT does w.r.t. bits requested, I haven’t looked at the code.
You are still calling through winex for one platform which might create the issue reguardless of what you decide to do.