I think that the simple answer to this question is that the present comparative methodology was largely established by Felsenstein 1985, American Naturalist. For mathematical convenience, he suggested Brownian motion as a null hypothesis, because "...the variance of the distribution of change of a branch is proportional to the length of time of the branch...", and then "...it is easy to see that the differences between pairs of tips... must be independent." Also: "...after one unit of time, the contrast [between a pair of tips] has expectation 0 and [easily defined] variance..."
He explicitly discusses whether Brownian motion is a reasonable model in the section "What if we lack an acceptable statistical model of character change?"
I would suggest reading that paper in more detail if you are interested in the details of Brownian motion applied to phylogenies.
A recent historical perspective on this influential paper can be found here.
For a more extensive bibliography/more details you can see here.