The fractal structure of the lung ends with relatively smooth tissue after a finite amount of branching. So the usual arguments about fractals notwithstanding, there actually is a finite area to discuss that makes sense.

Sure, but if you look closer you will see that all of the cells have holes for stuff to go in and out, and the proteins sticking up from the surface have shapes, and so on. That's why I said "the atomic scale" not "the microscopic scale."

Any claim about surface area that relies on the examination at the atomic scale is no longer due to the fractal nature of the lung's branches. Such claims would be equality true (or false) of non-fractal organs.

On the atomic scale surfaces are not continuous so they don’t really have a surface area. You can’t infinitely zoom in like you can in a real fractal.

That assumes the fractal nature continues at all scales, which won't hold here I suspect (i.e. it's not enough for the structure to exist on those scales, the scaling properties have to continue for the geometry to be "fractal")

You wouldn't have lung tissue at that point. You'd have atoms that came from lung tissue.