Specializations of elementary symmetric polynomials Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$
indeterminates. The $h^{th}$elementary symmetric polynomial is the
sum of all monomials with $h$ factors
\begin{eqnarray*}
e_{h}(\mathcal{S}_{x}) & = & \sum_{1\leqslant i_{1}<i_{2}<\ldots<i_{h}\leqslant n}x_{i_{1}}x_{i_{2}}\ldots x_{i_{h-1}}x_{i_{h}}
\end{eqnarray*}
which, from a generating function standpoint, can be built up as the
coefficients of the $h^{th}$ power of the following linear factorization
\begin{eqnarray*}
\prod_{i=1}^{n}(1+x_{i}z) & = & (1+x_{1}z)(1+x_{2}z)(1+x_{3}z)\ldots(1+x_{n}z)\\
 & = & \sum_{h=0}^{n}e_{h}(\mathcal{S}_{x})z^{h}
\end{eqnarray*}

Some usual specializations of the set $\mathcal{S}_{x}$ lead to
  known families of numbers and multiplicative identities: binomial
  coefficients for $x_{i}=1_{i}$, to $q$-binomial coefficients for
  $x_{i}=q^{i}$ and Stirling numbers of the first kind for $x_{i}=i$;

(i) For $\mathcal{S}_{1}=\{1_{1},1_{2},1_{3},\ldots,1_{n}\}$ 
\begin{eqnarray*}
(1+z)^{n} & = & (1+1_{1}z)(1+1_{2}z)(1+1_{3}z)\ldots(1+1_{n}z)\\
 & = & \sum_{h=0}^{n}{n \choose h}z^{h}
\end{eqnarray*}
we have binomial coefficients $e_{h}(\mathcal{S}_{1})={n \choose h}$
(ii) For $\mathcal{S}_{q^{i}}=\{q,q^{2},q^{3}\ldots,q^{n-1},q^{n}\}$
\begin{eqnarray*}
\prod_{i=1}^{n}(1+q^{i}z) & = & (1+q^{1}z)(1+q^{2}z)(1+q^{3}z)\ldots(1+q^{(n-1)}z)\\
 & = & \sum_{h=0}^{n}{n \choose h}_{q}q^{{h+1 \choose 2}}z^{h}
\end{eqnarray*}
we get the $q$-binomial coefficients (or Gaussian coefficients) $e_{h}(\mathcal{S}_{q^{i}})={n \choose h}_{q}q^{{h+1 \choose 2}}$
(iii) And for $\mathcal{S}_{i}=\{1,2,3,\ldots n-1\}$
\begin{eqnarray*}
\prod_{i=1}^{n-1}(1+iz) & = & (1+1z)(1+2z)(1+3z)\ldots(1+(n-1)z)\\
 & = & \sum_{h=0}^{n}\left[\begin{array}{c}
n\\
n-h
\end{array}\right]z^{h}
\end{eqnarray*}
the elementary symetric polynomial generates Stirling numbers of the
first kind $e_{h}(\mathcal{S}_{i})=\left[\begin{array}{c}
n\\
n-h
\end{array}\right]$

In this context, are there other specializations of the set $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$
  which lead to other families of numbers or identities?


You have mentioned what are known as the  (stable) principal specializations of the ring of symmetric functions $\Lambda$. If you haven't already, you should check out section 7.8 of Stanley's Enumerative Combinatorics Vol. II, where he summarizes the specialization you have have mentioned. In particular, using the specializations you have mentioned on the other bases for the symmetric functions i.e. the homogenous (complete) symmetric functions you can derive similar standard combinatorial formulas.
Stanley also mentions another interesting specialization for symmetric functions called the exponential specialization, which is the unital ring homomorphism $ex:\Lambda \rightarrow \mathbb{Q}[t],$ that acts on the basis of monomial symmetric functions by $m_\lambda\mapsto \frac{t^n}{n!}$ if and only if $\lambda=(1,1,\ldots,1)$ is the partition of $n$ consisting of all $1$'s, otherwise the map sends $m_\lambda$ to $0$. According to Stanley, this specialization is some sort of limiting case of the principal specialization.
The combinatorial significance of the $ex$ specialization as it allows one to prove that $$[x_1\cdots x_n]\left(\sum_\lambda s_\lambda(x_1,\ldots,x_n)\right)=e_2(n),$$ where $s_\lambda$ is the Schur function for a partition $\lambda$ and $e_2(n)$ is the number of involutions in the symmetric group $\mathfrak{S}_n$. This has some significance in the RSK correspondence.
P.S. I would make this a comment but I don't have enough reputation.
