No. That's like asking if Final Fantasy can keep achievements for Street Fighter. OpenBOR is not an emulator and it's not a game console. It is an engine you build games with, the same as Unreal or Unity. Each game is its own thing.
You can of course track achievements within a game all you like. Whatever you dream up and script.
For the record, it would be technically possible for two or more games to share data, but they would all have to be built specifically to do it and you'd need to know your way around the engine on a very advanced level to pull it off.