Do black metal vessels cook food more quickly than normal steel?

First, what you linked seems to be a pressure cooker. Pressure cookers cook completely differently from normal cookers, and yes, they are quicker, but not due to their color.

Now on to the actual question. The answer is: probably not. There are lots of factors involved, and the color isn't that important.

In baking, you get lots of infrared radiation cooking your food, especially in a small oven with uncovered elements (toaster oven). There, the color can have a significant effect on the rate at which your food is warmed, all other things being equal. Simple example: a steel pan and the same pan colored black. The steel pan will transfer more heat into the food in the same time period (even though the added dye layer already complicates it, because it reduces the effect somewhat).

But let's make the example more realistic. Steel pan vs. alumunium pan colored black. There, the material's thermal coefficient and its mass will be more important than the color. There are several factors acting in different directions here, so I can't tell you which pan will be able to output a specified amount of heat in shorter time (in fact, I suspect that if you wold plot the needed time against the specified amount of heat, the lines will cross somewhere). A really good explanation of everything important for understanding these factors is found here.

But your example wasn't for a cake pan, it was for a stovetop pot. And there, the color is even less important. On stovetop, the pan is in almost direct contact with a heating plate, and conduction plays a greater role in heating than radiation. The color only matters in radiation, not in conduction.

And last, let's say that you have somehow found a pan which outputs the same amount of heat in a shorter time only because of its color (this will probably be the black-dyed steel pot from my simple example). Does it mean that you will spend less time cooking? No, it doesn't. As a general rule, food which has been warmed gradually tastes much better than food which has been exposed to high heat. There are some exceptions (most notably, you don't get caramelization and Maillard below certain temperatures), but normally, you don't want to pump all the heat you have available into your food. You would end up with food which is scorched on the outside and raw on the inside. This is why most foods are prepared on a medium heat setting. And if you find a pan which transfers more heat to your food than usual, it doesn't mean that your food will cook faster. It means that you will have to turn down the heat to avoid burning your food. This can have its advantages - especially for your electricity bill - but it doesn't result in faster cooking.

If this is about the material, aluminuim has a much better thermal conductivity than steel, so it will heat up (and cool down) faster - thus making a difference in speed.

If this is about the color: this should have very little to no influence.