DFA and equivalence relation
Let me spell what $R$ being an equivalence relation means:

Reflexivity: for all $q \in Q$ there exists $a \in \Sigma$ such that $\delta(q,a) = \delta(q,a)$.
Symmetry: for all $q_1,q_2 \in Q$, if there exists $a \in \Sigma$ such that $\delta(q_1,a) = \delta(q_2,a)$ then there exists $b \in \Sigma$ such that $\delta(q_2,b) = \delta(q_1,b)$.
Transitivity: for all $q_1,q_2,q_3 \in Q$, if there exist $a,b \in \Sigma$ such that $\delta(q_1,a) = \delta(q_2,a)$ and $\delta(q_2,b) = \delta(q_3,b)$ then there exists $c \in \Sigma$ such that $\delta(q_1,c) = \delta(q_3,c)$.

Now it's no longer about DFAs. It's about functions $\delta\colon Q\times\Sigma \to Q$.