Yes wifi6 is backward compatible with other wifi versions.
The problem you have is all extenders like this use a feature called WDS to do this. WDS was not actually intended to be used for this purpose but most manufactures now implement it the same way even though there really is no official standard. Most routers have WDS so they should work with repeaters.
The "mesh" systems use another method and you can not mix different brands of mesh systems in most cases. But even mesh systems will support WDS to be compatible with older repeaters.
I would consider every other option before you buy any kind of repeater. This is where you will be happy to get any kind of signal rather than no signal. It also is not just plug the device in the remote room and you magically get good wifi like the marketing pretends.
The device must be placed where it can get a strong signal from the main router but still provide a good signal to the remote room. This is very tricky when you have walls and ceilings that absorb the signal. Lot of trial and error and in some houses that absorb lots of signal there is no good place.
Other options that I would try first assume you don't have ethernet. Moca is almost as good as ethernet but you need coax cable in both room. Then I would consider powerline networks. The ones with av1000 or av2000 numbers tend to work well in most houses but like wifi there are some houses they don't.