Let $L$ be a set of specific symbols and $\operatorname{Form}(S)$ be the set of all first-order formulas over $L$. Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ elementarily nonequivalent structures for $L$?
My reasoning is following: we know that $\operatorname{card}(\operatorname{Form}(L)) = \operatorname{card}(\operatorname{Sent}(L))$; for $L$-structures to be nonequivalent is to satisfy different sets of $L$-sentences and there are $2^{\operatorname{card}(\operatorname{Sent}(L))}$ sets of sentences which may differentiate $L$-structures.
Am I correct?
Your reasoning is correct. Here's a partial answer to the more interesting question in your comment. It's partial because I'm going to assume $L$ is countable.
A language $L$ has fewer than $2^{\aleph_0}$ classes of non-elementarily equivalent structures if and only if $L$ consists of finitely many constant symbols and finitely many unary relation symbols. In this case, $L$ has exactly $\aleph_0$ classes of non-elementarily equivalent structures.
First we'll argue right to left. Suppose $L = \{c_1,\dots,c_n, P_1,\dots,P_m\}$. The unary relations $P_i$ together partition the domain of an $L$-structure into $2^m$ definable subsets: for any subset $S\subseteq \{1,\dots,m\}$, we have the definable set $\phi_S(x): \bigwedge_{i\in S} P_i(x) \land \bigwedge_{i\notin S} \lnot P_i(x)$. Think of the $P_i$ dividing the domain of a model into a Venn diagram.
Now an $L$-structure is determined up to elementary equivalence by the following data:
1) For each $S\subseteq \{1,\dots,m\}$, the number of elements satisfying $\phi_S(x)$, among the countably many possibilities $0, 1, 2, \dots, \text{and "infinitely many"}$. By Lowenheim-Skolem, once a definable set has infinitely many elements, no sentence of first-order logic can distinguish between different cardinalities. Making this choice for each of the $2^m$ sets in the Venn diagram gives $(\aleph_0)^{2^m} = \aleph_0$ many possibilities. (If there are no relations, there is just one set in the Venn diagram, and this comes down to choosing the size of the structure.)
2) For each pair of constants $c_i$, $c_j$, whether $c_i = c_j$. These choices correspond to a partition of $\{c_1,\dots,c_n\}$, and there is some finite number of ways to do this, call it $B_n$.
3) For each constant $c_i$, which definable set $\phi_S$ contains $c_i$. There are $2^m$ possibilties for $c_i$, so in total there are at most $(2^m)^n$ many ways to assign the constants.
All in all there are at most $(\aleph_0)^{2^m}\cdot B_n\cdot (2^m)^n = \aleph_0$ many $L$-structures up to elementary equivalence.
Now we'll argue left to right. We need to show that if $L$ has
a) Infinitely many constant symbols, or
b) Infinitely many unary relation symbols, or
c) A single relation symbol of arity greater than $1$, or
d) A single function symbol,
then $L$ has $2^{\aleph_0}$ classes of non-elementarily equivalent structures. In each case we'll code up arbitrarily subsets of $\omega$.
a) Enumerate the constants $\{c^*\}\cup \{c_i\mid i\in \omega\}$. Given a subset $X\subseteq \omega$, consider a structure $M_X$ in which $c_i = c^*$ if and only if $i\in X$. Then $M_X$ and $M_Y$ are elementarily equivalent if and only if $X = Y$.
b) Similarly, enumerate the unary relations $\{P_i\mid i\in\omega\}$. Given a subset $X\subseteq \omega$, consider $M_X$ in which the interpretation of $P_i$ in $M$ is nonempty if and only if $i\in X$.
c) Say we have an $n$-ary relation $P(x_1,\dots,x_n)$, $n\geq 2$. We'll create structures in which the inputs $x_3,\dots,x_n$ of $P$ are irrelevant, so $P$ might as well be binary. Now the interpretation of $P$ on a structure $M$ will be as the edge relation of a graph. For each $n\in \omega$, there is a sentence $\phi_n$ asserting that $M$ contains a cycle of size $n$. So for each $X\subseteq \omega$, we can let $M_X$ be a structure consisting of $P$-cycles of size $n$ for each $n\in X$, and no other elements. If $X\neq Y$, then if $n\in X\setminus Y$, for example, we'll have $M_X\models \phi_n$ but $M_Y\not\models \phi_n$.
d) Similarly, say we have an $n$-ary function symbol $f(x_1,\dots,x_n)$. We'll create structures in which the inputs $x_2,\dots,x_n$ of $f$ are irrelevant, so $f$ might as well be unary. Now an $n$-cycle for $f$ is set of elements $\{a_1,\dots,a_n\}$ such that $f(a_i) = a_{i+1}$ for all $i<n$, and $f(a_n) = a_1$. Again, for each $X\subseteq \omega$, let $M_X$ be a structure consisting of $n$-cycles for $f$ for each $n\in X$ and no other elements.
In all of these examples, if there happen to be extra symbols in the language, we can interpret them arbitrarily. The point is that there are already sentences in the languages with just the symbols mentioned above that can distinguish $2^{\aleph_0}$ many classes of non-elementarily equivalent structures.
I encourage you to try to use similar reasoning to answer the question for uncountable $L$.