Is it meaningless to say $M\prec N$ for two proper class models? Kunen in page 88 of his "Set Theory" book says:

... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi (\overline{a})\Longleftrightarrow N\models\varphi (\overline{a})$) makes sense even for proper class models but $M\prec N$ does not, since one must quantify over all $\varphi$.



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*Why "quantifying over all formulas" is a problem of defining $M\prec N$ for proper class models but not for set models?

*Why the fact that we can't define $M\prec N$ for proper class models, doesn't produce any problem in dealing with proper class models of set theory? e.g. When we don't have a version of Lownheim-Skolem theorem for proper class models. 
 A: A good place to find details for what follows is the beginning of Kanamori's The higher infinite. The arguments below assume we are discussing (in) $\mathsf{ZF}$.
You cannot define a satisfaction predicate for proper classes in general. For instance, this would contradict Tarski's theorem on undefinability of truth. Roughly, satisfaction for $\Sigma_n$ statements is itself a $\Sigma_n$ predicate.  
For set models, the relevant quantifiers are bounded, so this issue is not present. In fact, books like Devlin's Constructibility explain in detail how satisfaction is formalizable in a $\Delta_1$ manner.
Having a way of formalize full elementarity between proper classes is not always that useful anyway: If $M,N$ are proper transitive class models of $\mathsf{ZF}$ and $M\prec N$ then $M=N$, since both $M,N$ agree on ordinals, and thus (for any $\alpha$) $V_\alpha^M$ must, by elementarity, coincide with $V_\alpha^N$.
On the other hand, there are ways to circumvent some of these issues. For instance:


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*Most arguments we care about only require $\Sigma_n$ elementarity for some (very) small $n$, and this can be defined with no problems.

*More useful than elementarity between models of $\mathsf{ZF}$ is being elementary embeddable. As before, to say that $j:M\to N$ is fully elementary would require defining a satisfaction predicate, so we cannot quite do this. But, for transitive proper class models of $\mathsf{ZF}$, if $j$ is $\Sigma_1$ elementary, then it is in fact $\Sigma_n$ elementary for all (standard) natural numbers $n$. So we discuss elementary embeddings (very common in set theory) by simply requiring that they be $\Sigma_1$ elementary.

*We cannot quite prove the Löwenheim–Skolem theorem for proper classes, but whenever we have a model (let's call it $V$ to simplify) of $\mathsf{ZFC}$, $V$ comes equipped with a natural hierarchy (the $V_\alpha$), and by reflection, we can find unboundedly many ordinals $\alpha$ such that $V_\alpha\prec_n V$. We can then form Skolem hulls of $V_\alpha$, and as long as $n$ is sufficiently large, this is as if we were taking elementary substructures of the full universe. This is actually a fairly used combinatorial technique. Many different hierarchies can be used in place of the $V_\alpha$, depending on context. 


All this being said, yes, we need a modicum of care when handling proper classes and proper class models. 
A: There is an issue here. Yes. The formulas $\varphi$ are not objects in the universe of set theory, they are objects in the meta-theory. So we cannot quantify over them internally.
If $\sf ZFC$ is to talk about itself, it can prove separately that for each $\varphi$, $M\prec_\varphi N$. But in order to have $M\prec N$ we have to go "one level up" to the meta-theory and prove it there. However, if we want $\sf ZFC$ to do the work, then what we really have is a schema of proofs (which can be translated into a single proof in the meta-theory, but that's not what we want).
When you talk about two set models, $M$ and $N$, then you really internalize the language into the universe of set theory. So now the formulas in the language are really sets, and you can quantify over them like you would over any other collection of sets.
For the second question, this is really repeating the same issue with why having inner models of $\sf ZFC$ doesn't prove the consistency of $\sf ZFC$. Proper classes are not objects in the universe of set theory (or rather, the universe of $\sf ZFC$), and the Lowenheim-Skolem, and other model theoretic notions, do not apply to them because these things only apply to objects inside the universe.
