This classy, if a tad expensive ($4,500!) sink doesn't come with fish, but it does offer a habitat for them. There's a light in the back as well as access ports on the top. The producer recommends that you get a professional plumber to install it.
Link -via My Modern Met
1) Oxygen exchange would be virtually non-existent. The surface movement created by the powerhead is pointless if the surface is not exposed to open air. Any fish you put in it would slowly suffocate to death over a period of days. If they even lived that long, because...
2) Tropical fish are extremely sensitive to rapid temperature changes and are only healthy within a very narrow band of a few degrees. There would be no effective way to regulate temperature in this "aquarium," since every time you used the sink it would act as a (forgive me) heat sink. The constant fluctuation in temperature would stress the fish, causing either disease and loss of appetite or killing them outright. But that may not have time to happen either, since...
3) When in a compressed water column with little or no cover, most tropical fish will go into hide mode. This is because they feel exposed to predators. They'd probably congregate in the corners (making it even harder to breathe in the already oxygen-poor water) and would likely refuse food, again leading to starvation, disease and eventual death.
As I said, a death trap. NEAT-or-AMA?