Comparing the variances of two distributions $X$ and $Y$ are discrete random variables with means $x$ and $y$ and variances $V_x$ and $V_y$, respectively.
Does the ratio $V_x / V_y$ have meaning, and, if so, what is it?
 A: Of course, if data are normal the variance ratio of independent random variables has Snedecor's F-distribution. And that distribution is widely used in checking assumptions and in ANOVA. 
For many discrete families of distributions the variance ratio will have a well-defined distribution that can be derived analytically or approximated by simulation. If the discrete distributions are approximated by normal, then one could check whether the variance ratio is approximately F. 
The main issue is whether there is a practical use for such a distribution. If you have an application in mind for particular discrete distributions, I suggest posting that.

I believe the only context in which I have seen a discussion whether discrete
distributions have equal variances (variance ratio equal to unity) is in some applications of ANOVA.
If data for $k$ treatment groups are Poisson counts or binomial counts, then
one may hope the counts are nearly normally distributed and try to use
a standard ANOVA to see if groups differ. The difficulty is that if the
groups have different population means then they are bound to have
different population variances, thus violating one of the assumptions
of a standard ANOVA design. (Example: For Poisson data population means
and variances are numerically equal.) 
This has given rise to a literature on 'variance stabilizing transformations'.
For example, one can show that if data are Poisson counts, then taking
square roots of the counts makes the variances nearly equal. Similarly,
for binomial counts, one divides by binomal $n$ to get proportions and
then takes the arcsine of the square root of each proportion to 'stabilize'
the variances among groups (that is, to make them nearly equal). While
these transformations do indeed stabilize variances, simulation studies
have cast some doubt on their usefulness in getting ANOVAs to make
correct inferences about differences among group means.
