With quatersawn white oak floors like you have, I'd go for a match. Quatersawn white oak is a fantastic floor. There isn't as much board movement or shrinkage and the rays in the wood really give it a unique look. Do NOT attempt to put in red oak next to it. It will look absolutely awful. The grain structures are totally different.
You can experiment a bit with different stains on some scraps. One thing I used to do to "age" the boards a bit is to apply a diluted coat of amber shellac before staining. It warms colors considerably lending to the illusion of patina. Experiment! If I were shooting at a match, I'd say try for a combination of Minwax Golden Oak, Provincial and amber shellac.

The photos indicate an old rift & quartersawn oak (although the 2 boards in the doorway may have been "patched in" because they appear to be plainsawn...
Hard to tell whether or not they are white or red oak by that photo... but I'd guess white oak.

As far as obtaining a "match"... we can probably do that, since custom milled flooring is our field of endeavor.
If you're interested, drop me a line at: david@selecthardwoodfloor.com with some of the details in regards to your existing floor.
Such as width, thickness, square footage needed, etc.
We can almost certainly help.

Pretty obvious that your floor is 1/2x2 white quarter sawn oak. One of the best floors EVER. I would absolutely match it. It would not be at all difficult. Install the other areas raw (unfinished) and then have it all sanded at the same time and stained to match. Matching an existing color is a lot harder than sanding all of it and then staining it all the same color. Matching is a science, but is still hit or miss, but a full refinish is sure to make your best possible match in color.

match because this one should be easy, and if not exact, be aware most older homes (in Minnesota anyway) have oak in the living areas and maple or pine in the bedrooms. Oak is always popular, and they even carry a fairly similar one at the big box stores. Do a transition of a different wood, width or color at the doorways to set it off!

What I hate seeing is when they go over old wood kitchen floors with CHEAP fake laminate, butting it up to wood in other rooms.

I'm not as quick to conclude that your flooring is "obviously 1/2 x 2" material, as my learned colleague Mr. Hennessy, who is familiar with working in Southern CA, where that was a predominant choice for many years.
There are factors such as YOUR location and the timeframe which would help determine the "history" of what's there.
As mentioned, the doorway appears to be plainsawn (& slightly wider) than the quartersawn material... which may indicate that it is 2-1/4" material... which is produced predominantly in 3/4" thickness...
If your house is 110 yrs. old, that will be a factor.
I'm not too sure how long 1/2" oak has been in production, but 25/32" (now 3/4") has been around a long time... which would probably be to your advantage in consideration of the fact that it will have been able to endure more refinishing's over the years than a 1/2" floor.

The best way to determine what you have is to STICK a thin ruler between the carpet & the wood floor & measure the thickness... and while you have that ruler in hand, measure the widths of the existing floor... then you'll have the information needed to find someone who can supply your product whether it's a "stock item" or would need to be replicated.

Have you considered cork? There are some great patterns and colors. You could find one that compliments your hardwood. I like cork in a family room as it is soft, quiet and never is cold.

Select floor co may be correct...I did not notice your location. I may have been a bit bold with my choice of words. I am familiar only with Californian standard sizes. I do still think it is 1/2x2 but it will be fun to find out. If our is 3/4 x 2 1/4 quarter sawn, then you have an even better floor than I thought.

Thank you everyone for your feedback. We live in central Illinois if that helps deciphering the wood type. My contractor is urging me to do an engineered hardwood which is easier to install but I'm leaning toward installing unfinished oak (if that is what type of wood we are dealing with) and staining to match. We refinished the original floors 7 years ago and they are holding up incredibly well. I'd rather not refinish them all over again just to match the new floor. Hopefully I can find a stain that looks very similar.

If you don't want to refinish them all once the new is installed, then you definitely should put a fresh topcoat on what was refinished 7 years ago in tandem with the final coat that goes onto your new flooring. Your color looks like either a golden brown (made by dura seal) or possibly a mixture of golden brown and quick dry (also a dura seal product). it should not be that difficult to get a close match even if you do not refinish all the floors, but a topcoat really should be done on the existing too for two reasons: 1. it will make the sheen throughout all the hardwood an exact match, and 2. It is a smart idea to re-topcoat your refinished hardwood floors about every 5 years. If you add topcoats every few years you will not have to do the entire refinishing process for a very long time. Make sure that if you go the matching route, that you get "quarter sawn" oak. It is a VERY different look than regularly cut oak, and the difference would be very noticeable.

Just FYI: When we built our addition our general contractor brought in a custom wood flooring specialist who first identified the species and cut of the old flooring in the rest of our house so our contractor could order it for the addition, then came back once it was installed, with a kit with many different stains, and experimented on sample boards with different colors and layers of stains until he felt we had a perfect match and I had approved it. We couldn't refinish all the adjoining old floors because they are thin enough after almost a century so that they have only one more sanding in them, and we didn't want to waste it. So they had to match the new floor to an existing stain and finish.