For example, I want to know the meaning(semantic) of "=" in FOL for the set theory. In structure $\mathscr{A}$, "$c_{1}=c_{2}$" is interpreted as $c_{1}^\mathscr{A}=c_{2}^\mathscr{A}$, i.e., the "=" in FOL for the set theory is interpreted as the = in meta language, which says "=" means =, and it does not make any sense. I still do not know the meaning of =.
Let's talk about the FOL with equality as it seems the one most logic books talk about. For FOL without equality, it seems easier to understand the meaning of "=" which is interpreted as a predicate.
The structure $\mathscr{A}$ has an underlying set $A$. Given two elements $a,b\in A$, it makes sense to ask whether $a = b$, i.e., whether $a$ and $b$ are the same element of $A$.
If you agree that this question makes sense, then there's no problem: we define $\models$ so that, for elements $a,b\in A$, we have $\mathscr{A}\models (a = b)$ if and only if $a = b$.
If you don't think this question makes sense, then you have a pretty nonstandard view of the nature of sets and their elements, which is going to make it hard to communicate about mathematics.