It's insanely hyperoptimized in about 11 million different ways. From massively increasing the surface area (iirc it's like over 100 m^(2) or something, for comparison your entire self is only like 2m^(2) on the outside, it's a real Tardis situation), to having ultra-thin diffusion membranes, to a specialized chemistry with an insanely high reaction constant that is then catalyzed even more so that it's beyond absurdly fast.

So yes the oxygen binds to the blood but no they basically aren't given time to bind, it's like a huge desperate crush of oxygen getting sucked into you at the absolutely fastest possible reaction rate that natural chemistry could evolve.