What is complexity class for #XOR-2-SAT?
The given problem is in $\mathsf {FL}$:

Solve if it is satisfiable. Let the result be number $s$. Can be done in logarithmic space.
Set $t=0$.
Count the number of clauses of form $(a\oplus \overline a)$ where $a$ does not appear in any other clause except clauses identical to that one. For each such clause increase $t$ by $1$.

Step 3 can be performed in log space as follows:
for every i in V:
    flagIndependent = true
    flagAppears = false
    for every j in C:
        if C_j isIdentical(`(x_i xor not(x_i))`):
            flagAppears = true
        elseif C_j contains(`x_i`):
            flagIndependent = false
    t += flagAppears * flagIndependent

Two variables i and j take space $\log(|V|)$ and $\log(|C|)$ respectively. Variables flagIndependent and flagAppears take $1$ bit each. Functions checking clauses also require $O(\log(n))$ space at most. Therefore the total space used is logarithmic.
The answer is $s2^t$.