Is it possible to replace this board in the floor with similar hardwood flooring?
I don't want to get into a long drawn out explanation on how to fix that plywood square. I will say that it is entirely possible to replace the plywood with a close match hardwood. The process is called "stitching". It involves removing the cut pieces of wood to bring it back to the original stagger then slip fitting the new wood back in while at the same time filling the hole left when you remove the plywood.

I expect that the hardwood flooring is 3/4" thick, likewise the plywood.  The plywood has to be removed and obviously replaced with a close match hardwood. There are a couple of very detailed segments on "Ask This Old House" that deal with stitching hardwood.  I'll try to find a link for ya later, but you can go to TOH and search hardwood floor stitching. It is a difficult job that requires a bit of skill and the patience of Jobe, but when done well, you can hardly tell it was a patch job

Found a video link for ya.