jmc

Since $\mathrm{lcm}[m,n]=108=2^2\cdot 3^3$, we know $m=2^a\cdot 3^b$ and $n=2^c\cdot 3^d$ for some positive integers $a$, $b$, $c$, and $d$. Furthermore, $\mathrm{lcm}[m,n]=\mathrm{lcm}[2^a\cdot 3^b,2^c\cdot 3^d]=2^{\max \{a,c\}}\cdot 3^{\max \{b,d\}}$, so $\max \{a,c\}=2$ and $\max \{b,d\}=3$.

Also, $\gcd (m,n)=2$, but $\gcd (m,n)=\gcd (2^a\cdot 3^b,2^c\cdot 3^d)=2^{\min \{a,c\}}\cdot 3^{\min \{b,d\}}$, so $\min \{a,c\}=1$ and $\min \{b,d\}=0$.

There are only 2 pairs $(a,c)$ that satisfy $\min \{a,c\}=1$ and $\max \{a,c\}=2$, namely $(1,2)$ and $(2,1)$. There are only 2 pairs $(b,d)$ that satisfy $\min \{b,d\}=0$ and $\max \{b,d\}=3$, namely $(0,3)$ and $(3,0)$. Therefore, there are $2\cdot 2=4$ possible quadruples $(a,b,c,d)$, so there are $\boxed{4}$ possible pairs $(m,n)$.