Assume that there is formula $\forall x : x \approx a$. Is this valid in first order logic?
Performing Hebrandization would suggest not with $c_1 \approx a$.
But then the Hebrand universe is just the set $\{a\}$. So if $x$ can only be $a$ this would be valid by the reflexivity of equality.
Is this just because Skolemization (and Hebrandization) extends the Hebrand universe? i.e the Hebrand universe is actually $\{a, c_1\}$.
Obviously, $\forall x (x=a)$ is not universally valid.
Consider e.g. a domain $D = \{ 0,1 \}$ and interpret the constant $a$ with e.g. $0$.
But the formula is satisfiable in every domain with a single element.
And this is what the Herbrand universe shows : the satisfiability of the formula.