You are going to have to sample several stains and combinations of stains, especially since your floors are cherry (with a custom color) and your staircase is oak. While it's not the same depth, I don't think the color looks bad on the oak...

Thanks! Yes headed out tonight to pick up samples. I think it needs a bit more warmth to go with the floors, since you can never get away from that undertone.

Your inspiration photo is handsome...but it has a much lighter floor and much lighter walls. And the light filled VAULT that highlights the wood is something you are missing. If you attempt a match like your the inspiration you are going to create a very dark looking space. The oak is very 'heavy'. That's a lot of trim that will be stained. And to get it to look good, you will have to go dark. Not medium....dark. Your floors are quite dark (we rank them as 'dark') which means your tone on tone match is going to be dark.

Are you leaving the houses trim WHITE? I would keep the trim and the stringers white. I like the idea of the hand rail matching or being DARKER than the flooring. Your newel posts could be white - same as the trim. Or BLACK.

Just be aware you have Red Oak. The red is going to be tough to get away from. You can use Jacobean to get rid of the red AND to darken the stain. But be careful with all the heavy trim work on the stairs. They are going to be powerful if you go dark. It will look very, very heavy.

You need someone to custom mix a stain that works with the floor and the stairs it could take a bit of work.

Make the wood white with dark balusters. I did that in my remodel and it looks great

Type in wood stains under photos of the site and see "Modern Manor". A dark grey wood stain with your iron on hand rails and balusters would look great as the blackish grey recedes and will not look as bulky and keep your trim white to match the rest of the house. Should be stunning though a bit more modern than your inspiration photo.

I would go lighter than the floors on the wood posts and hand rails. I think just special walnut on its own would blend and it would not make the house feel too heavy or dark. White paint on them may be to stark given the ornamental metal railing you chose. If it’s in the budget I would change the carpet on the stairs to wood treads that match the floor and paint the risers white.
I did that with the stairs in our house and it immediately made the house look more elegant.

I would sand and stain what I have to a cherry finish to complement the floors. There will be less grainy movement than oak.