Well, in the given answer here https://social.msdn.microsoft.com/Forums/vstudio/en-US/ee3df896-1d33-451b-a8a3-716294b44b2b/socket-programming-on-64bit-machine?forum=vclanguage there is written:
The implementation is in a file called ws2_32.dll and there are 32-bit and 64-bit versions of the DLL in 64-bit Windows.
So the build as described in my question is correct.
The C#-Project has to be explicitly set to x64.
AnyCPU won't work and throw the error shown in the question above.
Everything is working now. I'm leaving the question and answer as this is a full explanation of how to get go-code running out of C#.