I assume, that we all have (beside any set-thoretic background) a good intuition, what "the usual natural numbers" are, although we seem not to be able to describe them precisely.
So having said this: If we have a ZFC-universe $U$ it makes sense to seperate the members of $\omega_U$ absolutely into standard "numbers" (those members of $\omega_U$, which represent a "usual natural number") and non-standard "numbers".
Furthermore I want to assume the consistency of ZFC.
My Question is:
Does any ZFC-Universe $U$ "exist", for which $\omega_U$ does not contain non-standard "numbers"?
If you assume there is a "true" universe of sets, the answer is yes, but of course this is not a mathematical argument. If you refer to set models, the assumption that there is such a model (an $\omega$-model) is strictly stronger than the assumption that set theory is consistent, which is itself not provable in set theory (by the second incompleteness theorem).
Recall that $\mathrm{Con}(T)$ is an arithmetic (in fact $\Pi^0_1$) statement whenever $T$ is recursively enumerable; for example, $\mathrm{Con}(\mathsf{ZFC})$ is the statement "no $n$ codes a formal proof of $0=1$ from the axioms of $\mathsf{ZFC}$". If there is a model of set theory, this statement is true (in the standard natural numbers), and therefore any $\omega$-model of set theory is also a model of $\mathrm{Con}(\mathsf{ZFC})$, of $\mathrm{Con}(\mathsf{ZFC}+\mathrm{Con}(\mathsf{ZFC}))$, etc, so the consistency strength of "$\mathsf{ZFC}+$ There is an $\omega$-model of $\mathsf{ZFC}$" not only is higher than that of $\mathsf{ZFC}$, it is in fact much higher. This theory cannot prove its own consistency, so (if there are $\omega$-models at all) there are $\omega$-models of $\mathsf{ZFC}$ where no set model of $\mathsf{ZFC}$ is an $\omega$-model.
The nicest models of set theory not only are $\omega$-models, they are in fact well-founded (sometimes called $\beta$-models). Any well-founded model of set theory is also a model of "there is an $\omega$-model" (this follows from Mostowski's absoluteness theorem, a statement of which you can see in this answer on MO), so assuming the existence of well-founded models is a further (significant) jump in consistency strength. That said, most set theorists routinely assume (believe?) there are such models, and much more.