Not exactly. Yes, the "dimension" in dimensional analysis refers to the powers, and the "dimension" in "spatial dimension" refers to the directions. In this sense, it is two meanings for one word.
(I'm using "degree of freedom" in a loose sense here)
But, if you look at the more basic meaning of the word--"a 'degree of freedom'", then, both fall within this meaning. Measurable systems are many times classified by saying that the system is "N dimensional", where N refers to the number of variables present. For a point object at rest in space, we have 3 spatial DOFs--so the universe is "3 dimensional" (time can be added to the mix easily, and, less easily, you can talk about adding more spatial dimensions in String Theory).
Similarly, mass, length, time, current, etc can also be used to classify data, giving rise to similar set of degrees of freedom.