Modern x86-64 processors are still capable of running 16-bit code but, assuming you want to do it from within a modern operating system, it needs to be 16-bit protected-mode code, which isn't that useful since I think most 16-bit games run in real mode. Hence why DOSbox/Bochs are used instead (also, for emulating the original sound/graphics hardware).
AFAIK, no 64-bit version of Windows ever shipped with native 16-bit support, though. That means 16-bit support on x86-64 in practice was only ever usable on a 32-bit operating system, which means you still couldn't have 64-bit, 32-bit, and 16-bit code running side-by-side, even if the hardware could theoretically support it.
Intel does want to get rid of most of the compatibility modes, including all 16-bit support, but they haven't done it yet: https://www.intel.com/content/www/us/en/developer/articles/t...
