In this paper, we show for generalized $M$-ary search trees that the Steiner distance of $p$ randomly chosen nodes in random search trees is asymptotically normally distributed. The special case $p=2$ shows, in particular, that the distribution of the distance between two randomly chosen nodes is asymptotically Gaussian. In the presented generating functions approach, we consider first the size of the ancestor-tree of $p$ randomly chosen nodes. From the obtained Gaussian limiting distribution for this parameter, we deduce the result for the Steiner distance. Since the size of the ancestor-tree is essentially the same as the number of passes in the (generalized) Multiple Quickselect algorithm, the limiting distribution result also holds for this parameter.