In some approaches to first-order logic based on cylindric algebras -- e.g., the treatment in Appendix C of Blok and Pigozzi's 1989 Algebraizable Logics as well as the systems discussed in sections 7.5--7.9 of Andréka, Németi, and Sain's "Algebraic Logic" (2001) -- identity is treated as a primitive logical constant symbol rather than a binary predicate and the quantifiers are sequences of primitive unary connectives (a pair of quantifiers for each variable). For instance, Blok and Pigozzi write (p. 68):
[The language] also contains a primitive nullary (constant) symbol $v_i = v_j$ for each [pair of variables in the language] $v_i, v_j$. Each of the symbol complexes $\forall v_i$, $\exists v_i$, and $v_i = v_j$ is to be considered indivisible.... It would be better from a logical point of view to denote these connectives by something like $\forall_i, \exists_i, \text{and} =_{ij}$....
Am I right to think that this means that in, e.g., the axiom $\exists v_i (v_i = v_j)$ the quantifier isn't binding $v_i$ in the identity because, in a sense, there is no variable to be bound? More generally, how is variable binding to be understood in these approaches?
The elements of a cylindric algebra with $n$ variables are thought of as $n$-ary relations on a base set $U$.
The equality $v_i=v_j$ is then the diagonal relation $\{(u_1,\dots, u_n) : u_i=u_j\}$, a constant element of the algebra.
The quantification corresponds to cylindrification: we 'project' the relation in hand to the other $n-1$ dimensions then pull a cylinder above it along the whole $U$ in the chosen dimension, $$\exists v_i:R\ =\ \{(u_1,\dots, u_n) :\exists v_i. (u_1,\dots, v_i, \dots, u_n) \in R\} $$ Then $\exists v_i:v_i=v_j$ is the full relation $U^n$.
In a sense, this formalism simply steps over the question of bounded variables, as all elements of the cylindric algebra are treated equally, moreover it enables us to 'reuse' already bound variables even within the bounded subformula.
Nevertheless, one can recognize the 'independent' variables for any element: namely, element $R$ is independent from $v_i$ if $R=\exists v_i. R$.