Just an addendum to Ricky's answer: the multiplicity is indeed 1 which can be proved as follows.

An occurrence of ${\rm det}(V)^{\otimes 2}$ inside $({\rm Sym}^2(V))^{\otimes n}$ is the same thing as a nonzero joint multilinear ${SL}_n$-invariant of $n$ quadratic forms $Q^{(1)},\ldots,Q^{(n)}$ in $n$ variables. By the first fundamental theorem of classical invariant theory, this must be a linear combination of expressions (after choice of coordinates) of the form
$$
\sum_{i_1,\ldots, i_{2n}=1}^{n}
\epsilon_{i_1,\ldots,i_n}\ \epsilon_{i_{n+1},\ldots,i_{2n}}
\ Q_{i_{\sigma(1)},i_{\sigma(2)}}^{(1)}
\ Q_{i_{\sigma(3)},i_{\sigma(4)}}^{(2)}
\cdots
\ Q_{i_{\sigma(2n-1)},i_{\sigma(2n)}}^{(n)}
$$
where $\sigma$ is a permutation of $\{1,\ldots,2n\}$.

Here the $Q_{i,j}^{(a)}$ denote the matrix elements of the quadratic forms and $\epsilon_{i_1,\ldots,i_n}$ is completely antisymmetric with the normalization $\epsilon_{1,\ldots,n}=1$.

If you take a symmetric matrix $A$ and a skew-symmetric $B$ then $\sum_{i,j}A_{ij}B_{ij}=0$ because you are contracting two symmetic indices with two antisymmetric ones. The same is true if $A$ and $B$ are tensors with more indices that are frozen.
Thus the above expression is zero for all permutations $\sigma$ which send two elements of the same block of the partition $\{\{1,\ldots,n\},\{n+1,\ldots,2n\}\}$ to the same block of the partition $\{\{1,2\},\{3,4\},\ldots\{2n-1,2n\}\}$.
It is then easy to see that all you get are multiples of the expression corresponding to say the permutation $\sigma$
defined by
$$
\sigma(i)=2i-1\ \ ,\ \ \sigma(n+i)=2i
$$
for $1\le i\le n$.
Moreover, this invariant is not zero because when specializing to all
quadratics being equal to say $Q$ this gives the non identically vanishing polynomial $n!\ {\rm det}(Q)$.

Note that the above permutation $\sigma$ is not the only that works. There are $2^n\ n!^2$ permutations which satisfy the combinatorial requirement I mentioned but their corresponding invariants differ by a $\pm 1$ factor.

Finally, as remarked by Darij, this easily generalizes to occurrences of ${\rm det}(V)^{\otimes k}$ inside $({\rm Sym}^k(V))^{\otimes n}$.