In Zig the default target is native, which turns on all the applicable CPU features. If you want to target something other than the native machine you can use --target-os --target-arch parameters. I haven't exposed extra CPU options for cross compiling yet.
Do you consider the default architecture targetted by GCC (e.g. some old Intel) to be cross compiling? That is, can I make a binary that supports most x86 processors rather than just those with the particular extensions I support, with the current Zig compiler?
Yes. You can pass the native OS and native arch as the cross compilation target, and produce a binary that runs on your machine and others that don't have the same extra CPU features.