The only real solution is to strip the patch by using a scraper and
stain to match . You can go to the expense to replace it with red oak if
that is what you have but staining will still be likely necessary making
replacement unnecessary. I have matched stain many times, new oak to old
oak, pine to oaks to walnut etc etc. It will take someone with multiple
stain colors, experience and patience. Once the stain has cured a final
color check can be done by wetting the wood with say thinner, standing
back and compare results. Finish the patch by brushing in an equal
amount of coats you have now on the floor then recoat the entire floor.

It is time consuming to match stain, inbetween dry times and color
changes it would take me maybe 3-5 trips to stain it, and 3-5 to finish
it. or 1-2 weeks of an hour a day, then final coating, not cheap or
easy, but it is your house and should be done right.

A pro would have known this before hand, and more then I have
mentioned.

You contracted for a match, that is what I would pay for. You should
call out a real experienced pro to see it, as we can`t, someone who
knows how to do a close color match.