Register renaming is a microarchitecture feature; your OS can be completely oblivious to them because it has no way of using (or even knowing in many cases) them. Register windows, OTOH, require the OS to be aware of the “spill stack”. The difference is also in their purpose: register renaming is for OOO superscalar execution pipes to allow multiple operations at once; register windows are an attempt to avoid pushing “clobbered” registers to the stack when calling functions.

Nothing precludes an architecture from having both: register windows to avoid register clobbering and register renaming to allow dozens/hundreds of micro-ops “in flight” at the same time.