Define a standard bitableau of size $n$ to be a pair $(P_1, P_2)$ of standard tableaux of total size $n$ such that each of the integers $1,\dotsc, n$ occurs exactly once in either tableau.
Then $I_2(n)$ is the number of pairs of standard bitableaux $((P_1, P_2), (Q_1, Q_2))$ of size $n$ such that $P_1$ has the same number of cells as $Q_1$. In fact, the $j$th summand in the sum defining $I_2(n)$ is the number of such pairs where $P_1$ and $Q_1$ have $j$ cells.