Objective C is just as open as Swift is I think

Swift is FLOSS, Apache Licensed and built and collaborated on GitHub: https://github.com/apple/swift

Objective C has not had an official open source release, so far as I know.

ETA: Arguing of course how close the gcc/clang implementations of Objective C may or may not be to the Xcode compilers, of course.

Xcode uses Clang/LLVM to compile Objective-C (and used gcc before adopting LLVM). To my knowledge there's nothing closed source involved in the compiler. I had Clang/LLVM compiling Objective-C on Linux a long time ago. Visual Studio for Windows has been able to compile Objective-C for about a year (see https://developer.microsoft.com/en-us/windows/bridges/ios).

Of what significance is Swift's cross platform support here? Objective-C has had cross-platform support for decades.