"How can we match the stains?"

It require's a skilled and experienced individual to match stain hands on. If one could look at pictures then mix and send matching stain to customers,Sherwin Williams would pay them handsomely for their services. Try and get your hands on a color chart to compare with stairs. Chose 2 additional colors that look like they might mix with the stain you already have. Buy some 1 oz disposable clear plastic medicine cups at drug store. Measure and mix small batches until you get a match. When you think you have a match and stain is dry,apply clear to sample and let it dry before moving forward.

But the real question is will you care in six months. I don't think so. Yes, I can see that the color is redder. However, in the second pic with more shade, I can't see the difference and it isn't huge anyway. A stairway isn't a focal point. Also, do you want to further risk getting stain on those beautiful rug runners?

The only thing that bothers me in the photos is the quarter round 'shoe mould' in the bottom photo. Looks totally afterthought.

Different woods are supposed to be different colors.

At this point, trying to match the pine stain to the oak is pretty much impossible. Best bet, if the color diff is realllly, reallly bothering you is to pull the carpet on the stairs, paint & reinstall the carpet.