If I’m understanding this correctly, the area between the two sets of stairs is this discontinued floor, the both sets of stairs are carpeted and the main living area (up the stairs) is also the discontinued flooring. Is this correct?

If so, eliminate the discontinued flooring in between the two sets of stairs and replace it with whatever you choose to do for the stairs. You can keep the flooring upstairs as it is, but changing the floor at the entry to match the stairs will make it less disjointed and more intentional.

Do not try to match it too hard or it will look just like you tried to match it. I would purposely contrast, but it needs to compliment. If you pull a color from the current floor, I would lean more toward light (although hard to tell, I can make out some very subtle light streaks). I would steer clear of the darker tones as dark floors tend to show everything and these sets of stairs happen to be an integral and unavoidable part of the house, which will make the high traffic very conspicuous on dark floors.

I also suggest rugs!

Do not try to match. It wont look right. Its better to contrast the floors than have a half match. White Oak makes excellent stair treads. Its durable and takes a custom color well. I saw another thread on Houzz last week where someone had Poplar treads that were being dinged up. Make sure you pick a hard species like Oak.