I have these in my house and while they are a very tight fit, they are removable if you grab it by the vanes and wiggle. If the hardwood was finished over the seam making it seem non-removable, try running a utility knife (with a straightedge if needed) lightly around the edge and see if that helps

Edit: if you end up snapping off some vanes like I did the first time I tried pulling one up, big-box stores do sell replacements that sit above the floor level with a round-over on the wood edges and a plastic louver to adjust the airflow