Is this extension of ZFC known to be outright inconsistent? Is the following first-order theory known to be outright inconsistent? Adjoin to ZFC a unary function $U$ together with the following axioms. 1. If $\alpha$ is an ordinal, then there exists an (uncountable) strongly inaccessible cardinal $\kappa$ such that $U_\alpha = V_\kappa.$ 2. If $\alpha$ and $\beta$ are ordinals satisfying $\alpha<\beta$, then $U_\alpha \in U_\beta$. 3. (Schema; for all sentences $\varphi$ in the first-order language of $\in$): If $\alpha$ is an ordinal, then $\varphi$ holds iff $\varphi$ relativized to $U_\alpha$ holds.

No, this is not inconsistent, at least not relative to the existence of a Mahlo cardinal.

Suppose that $\kappa$ is a Mahlo cardinal, then $V_\kappa$ has a club of ordinals $\alpha$ such that $V_\alpha\prec V_\kappa$. Therefore there is a stationary set of inaccessible cardinals satisfying this.

Simply enumerate these inaccessible cardinals and let $\alpha$ be the enumeration of this stationary set. Then $U_\alpha=V_\mu$ for some inaccessible $\mu$ and $V_\mu\prec V_\kappa$.

Now cut the universe at $V_\kappa$, and this satisfies this extension of $\sf ZFC$ that you suggest. You can probably get away with much less than a Mahlo.