The speed of light has the dimension distance/time. Distance is a relative measure of space (length, height, depth), time is a measure of... time. When we talk about relativity, the distinction between space and time isn't very useful, so those are actually the same kind of measure. So distance/time is the same thing as distance/distance or time/time or 1.

I suspect this is precisely the half truths and cargo cutting you're referring to. Maybe you could explain what the problem here is.