Speed of light is roughly 186 000 miles per second, and roughly 300 000 kilometers per second, and as it turns out, 186 000 is very much different from 300 000. The beauty of fine structure constant is that it comes out to be the same numerical value regardless of what units you choose for your measurement.
The speed of light has units. The fine structure constant is a dimensionless quantity, like pi.
In special relativity and electrodynamics c is sometimes used as a dimensionless constant (equal to 1).
In those subjects they choose different units so that the speed of light is 1 of that unit, but it still has that unit associated to it. So it is not dimensionless.
The reason it is indeed left dimensionless is that in formulas it is impossible to keep track of the dimension of an implied factor that is equal to 1. In d = t, therefore, distance and time are actually measured in the same units (usually of time).