Your stairs look warm. This is just a guess, but some types of wood yellow with age and UV exposure, especially if they were sealed with urethane that's not water based. You might be better off sanding the stairs down and then staining/sealing them with the same product that was used for the new floors. That way they can all age together. Water based topcoats don't yellow like the urethane ones do.

Or if your budget is really tight, then you might need to settle with a match that's close but not perfect, then put a runner on your stairs so only a little bit of wood is showing.

Or paint the stairs, and then you can choose literally any color of floor you want!

Don't paint those stairs!

I'd suggest having a flooring professional come in for a consultation. You'll have more choices with a site-finished floor and those people can match anything. Well, they can't match that fabulous oak, but they can mix up something that will look good with it.

Darken the stain on the new flooring to match the existing. Do not paint the stairs.