basic_private.h should be added here.

Is a private struct for basic_shaders.c no need to add it to a header file.

The interface of the *initialize* functions are a bit confusing.
It creates,compiles the shader, update the sh_data and returns the newly created instance.

Would rather use BASIC_shader_create_depth_sh that only creates, compiles and returns the newly shader.
The caller would be responsible to update the sh_data.

BasicEngine doesn't use a library yet (see image_shader.cc as example)
The *library_ensure functions is used to initialize a DRWShaderLibrary. Where the libs (here view_lib&pointcloud_lib) is added.
The shaders can then be constructed using DRW_shader_create_with_shaderlib

Perhaps it is better to remove the library_ensure function or implement this in a separate patch (only one change per patch).

We use a lazy initialization of shaders as done in the BASIC_shader_*_sh_get functions. no need to do this at this point.

Can use something like:

GPUShader **sh_data_as_array = (GPUShader **)sh_data;
for (int i = 0; i < (sizeof(BASIC_Shaders) / sizeof(GPUShader *)); i++) {
    DRW_SHADER_FREE_SAFE(sh_data_as_array[i]);
}

would call the int i and use static_cast to cast to the enum. Seems like this could also be done for test_overlay_glsl_shaders in a separate patch.