Do all equilibria in 2 player zero sum games have the same distribution over outcomes I know that in a 2 player zero sum game all equilibria give each player the same expected value, but is it the case that they also induce the exact same distribution over payoffs?
Or might there be higher and lower variance equilibria with the same expected value?

You've got the right idea. As a further example, consider what you get when you cram two matching pennies games together as follows. There's obviously one mixed equilibrium across A/a and B/b and another across C/c and D/d, both of which have the same expected payoff but different variances.
\begin{array}{ccccc}
& a & b & c & d \\
A & 1,-1 & -1,1 & 0,0 & 0,0 \\
B & -1,1 & 1,-1 & 0,0 & 0,0 \\
C & 0,0 & 0,0 & 2,-2 & -2,2 \\
D & 0,0 & 0,0 & -2,2 & 2,-2
\end{array}
