As @cyanfish mentioned in comment section, "it's vital that the bottom CYGTLS_PADSIZE bytes of the stack are not in use before you call cygwin_dll_init()". I have tried to reserve those bytes via stackalloc and it seems to work.

IntPtr pcygwin = LoadLibrary("cygwin1.dll");
IntPtr pcyginit = GetProcAddress(pcygwin, "cygwin_dll_init");
Action init = (Action)Marshal.GetDelegateForFunctionPointer(pcyginit, typeof(Action));

var thread = new Thread(() => {
    unsafe {
        var bytes = stackalloc char[8096];
        init();
        
        // Now I can freely call functions from cygwin1.dll
    }
}, Int.MaxValue);

thread.Start();