If your API is c++ based, you ship a static lib. If the game engine is "closed" source, you also a have static lib (or a set of) for the game engine too, and "exporting" is actually linking (basically a binutils ld invokation). Ofc, you better use the same c++ compiler _VERSION_ than the game engine devs did use, or you are good for c++ hell.

If your API is C based (or core platform ABI), and you should design such API for interop with any framework anyway, you can ship a shared lib (but it has to statically load only the common glibc libs, and that with "old" symbol versions, and the other things too, see my previous post).