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Let's say that I have one population that can be divided by a 2-level factor. I run MDS twice (using prcomp() {stats} in R), using the 2-level factor to separate my subjects.
If I reduce to two dimensions, I now have two plots. Let's say that on each of the MDS plots, I have 5 points (with the same labels). I can calculate a total of 10 (5x4/2) distances for plot 1, and 10 distances for plot 2.
Is it meaningful to quantitatively compare the distances of plot 1 and plot 2?
Maybe, but you've said nothing about why you've chosen MDS in the first place. MDS is useful for visualizing similarities between objects but, in my opinion, doesn't easily lend itself to much deeper analyses.
For me, an easier, more straightforward and meaningful comparison would be to run a MANOVA with the labelled features as dependent variables and the 2-level factor as the independent variable. Typically, this would be followed with a canonical summary of the predictions. Then, run the standard MANOVA statistics, e.g., Wilk's Lambda, Roy's g.c.r, etc., on that.
A good intro to this technique is described in this 2014 paper by Warne A Primer on Multivariate Analysis of Variance (MANOVA) for Behavioral Scientists.
If you simply run MDS twice on the same data set, then the two results should be equivalent, i.e., they may not look identical due to different random initializations, but they should represent the same thing. So I'd say that, in general, no, there is no obvious reason to compare the distances between two MDS plots of the same exact data set.
