higher focal length affects your image because you have to move further away from your subject. The background will then seem closer to the subject.
That's only indirectly true. What you're really describing there is a lens with a narrow field of view (a.k.a., a "telephoto lens.")
I also understand that larger sensors require one to get closer to their subject in order to fill the frame.
The field of view is related to both the focal length and the sensor size. For any given sensor size, lenses with a longer focal length will give you a narrower field of view, and lenses with a shorter focal length will give you a wider field of view.
For a lens of a given focal length, if you put a smaller sensor behind it, the sensor will see only a small part of the whole image (i.e., a narrow field of view), whereas a larger sensor in the same place will see a wider field.*
Does that mean that a smaller sensor would be better to reproduce this style?
A smaller sensor is better when you want a smaller, lighter, less expensive camera that uses smaller, lighter, less expensive lenses. A bigger sensor is better in most other ways.
... this style...
I don't see much in common between the two pictures; They're both black and white, both architectural subjects, both shot with the camera pointing perpendicular to a flat wall.
The first one—the commercial building—Appears to be lit by bright sunlight at a shallow angle to the wall, but it has an "underexposed" look to it. Can't tell if it actually was underexposed in the camera, or if that's a deliberate low-key effect (i.e., no white pixels anywhere within it) that might have been done in post production. It's hard to say much about the camera's angle-of-view because the subject has very little depth.
In the second picture—the residential one—it's the lighting, not the subject, that is very "flat." Looks like it was taken under an overcast sky, although there is just a hint of shadows of the bushes on the driveway. It looks like a print on "high contrast" paper, partly because the film grain is so visible, and partly because of the featureless-blackness of the bushes.
I would guess that the lens used in the second one had a "normal" field of view (i.e, neither telephoto, nor wide-angle.)
* Up to a point. A lens that is purpose-built to be a telephoto lens for a small sensor is unlikely to actually work as a wide angle lens for a larger sensor because it will be designed to deliberately exclude rays of light that come from outside of its intended field of view. Excluding rays that won't hit the sensor is desirable in any lens, because it will reduce lens flare and improve the contrast of the pictures.