Because a model of a first order theory is not allowed to use a proper class as its domain, we can't use the universe of the set theory from the "meta-level" directly as a model for a first order theory of its axioms.
This situation alone already appears paradoxical to me, but it gets worse if I look at some different set theories I could use at the "meta-level". If I use NFU, there seems to be no problem to use it directly as a model of its axioms. (However, I'm not very familiar with NFU, so I could be wrong.) If I use ZFC or NBG on the other hand, it looks like they are unusable as a model of their own axioms. Allowing to use a class as the domain for a model also seems problematic, because relations are modeled as sets in ZFC (so for example $\in$ is not a relation in the sense of ZFC). The situation gets even more strange if I use pocket set theory, because then any set theory (including pocket set theory itself) seems to have an inner model (if the axiom of free construction is added) in that set theory (which is necessarily a countable model). However, I have the impression that it would be even possible to allow the use of a class as domain for a model in case pocket set theory is used at the "meta-level". (However, I'm not very familiar with pocket set theory, so I could be wrong.)
Is what I described above considered to be a paradox? Is there a "canonical" resolution of that paradox for ZFC (or NBG). Is it really the case that this paradox only happens to a lesser extent for NFU and pocket set theory, or is it just my unfamiliarity with these theories that prevent me from noticing that the same paradox also applies to these theories?
"Because a model of a first order theory is not allowed to use a proper class as its domain, we can't use the universe of the set theory from the "meta-level" directly as a model for a first order theory of its axioms."
Proper classes are used as models in the metatheory especially in relative consistency proofs. Proper class models provide finitistic proofs in the metatheory.
For example, the proper class model $V = \{x : x = x\}$ gives you the finistic proof in the metatheory that $\text{Con}(ZFC) \Rightarrow \text{Con}(ZFC)$, which is absolutely trival.
Some less trivial example of class models, let $ZF^-$ denote $ZFC - \text{foundation}$. The proper class $WF$ of well-founded sets is a proper class model satisfying the axiom of foundation. Hence $\text{Con}(ZF^-) \Rightarrow \text{Con}(ZF)$.
Also letting $L$ be Godel's Constructible universe, $L$ is proper class model of $ZF$, $V = L$, $AC$, and $GCH$. This proper class model yield the finitistic proof of relative consistency of these axioms from consistency of $ZF$.
So at the "meta-level", proper class models are useful for providing consistency proof in the metatheory. The formal precise logic of class models involve relativization.
However, sufficiently strong set theory is not capable of proving that there is a model of itself. The existence of a model of $ZFC$ proves consistency of $ZFC$. Hence $ZFC \vdash \text{Con}(ZFC)$. By the incompleteness theorem, this can only occur if $ZFC$ is inconsistent.
$ZFC \vdash Con(X)$ is stronger than the finitistic $\text{Con}(ZFC) \Rightarrow Con(X)$, where $X$ is some theory.