I will give an example to quickly introduce you to my problem. Lets say I have a language L = {p}, where p is an unary predicate symbol, and an interpretation M = {D, p - "x is an even number"}, where $D$ is a universe with elements D = {2, 3, 4, 5}. Also I denote that the predicate $p$ is true in this interpretation for elements {2, 4}.
Now lets say I have two formulas $(\forall x) p(x)$ and $(\exists x) p(x)$. First formula is obviously false in this interpretation $M$ and the second formula is obviously true (because an element $2$ and $4$ is said to cause a formula $p(x)$ to be true in this interpretation $M$ and therefore there exists an element $x$).
To this point everything is clear to me because those examples with finite domains are shown as an example of what it means for a formula to be true in most textbooks for mathematical logic. Lets get to the point where it is not clear to me because no textbooks cover the following (or I didn't find out any). For example instead of the domain D = {2,3,4,5} we will have an infinite domain $N$ of all natural numbers. Since it is an infinite domain, I cannot explicitly write all the elements of that domain. Now the problem is: How do I declare the same way as before that a predicate $p$ is true only for all the even numbers in the domain $N$? Since it is an infinite set, I cannot write all the elements for which $p$ is true. I could somehow shorten it and write that $p$ is true for elements {2,4,6,8,10,12, etc.}. However for this method some common sense of what it means to be an even number is necessary. My question is: How do we write in FOL that a formula is true for infinitely many elements such as even numbers in the domain $N$ without relying on a person's knowledge of what even number is? In textbooks when it comes to these examples with large or infinite domains, they no longer clarify why it is the case that number 4 in a predicate $p$ causes that formula to be true. It seems to me as if they were only relying on some common sense of a person.
The model theory here is a red herring; this is really an issue with how we describe infinite sets at all. E.g. when I say "$\mathbb{N}$," how do you know what I mean?
Ultimately, the most "universally satisfying" answer in my opinion is that we embed everything in a "large" formal system (like ZFC). When we talk about "the natural numbers," what we're really doing "under the hood" is manipulating certain finite strings of symbols (= well-formed formulas in the language of set theory) according to well-defined rules (= the ZFC axioms and rules of first-order logic). This translation is extremely tedious, but automatic.
For example, there is a natural translation of each of the following statements into the language of set theory:
There is a unique set $x$ such that $\emptyset\in x$, for each $y\in x$ we have $y\cup\{y\}\in x$, and for each $z$ with the previous two properties we have $x\subseteq z$.
With $x$ as above, there is a unique set $e\subseteq x$ such that $\emptyset\in e$ and for all $y\in x$ we have $y\in e\leftrightarrow y\cup\{y\}\not\in e$.
The ordered pair $\langle x,e\rangle$ exists (where $x,e$ are as above).
Each of these sentences can be proved in ZFC, and ZFC-proofs about $\langle x,e\rangle$ are the "formal counterpart" to our natural-language "rigorous proofs" of statements about the structure $(\mathbb{N};\{$evens$\})$. More complicated notions are more difficult to translate - e.g. "$\models$" is a real hassle - but the brute-force-massive-ignorance approach always (if painfully) gets the job done.
Put another way, in this approach "naive" reasoning about mathematical objects is taken to be a proxy for concrete manipulations of finite strings. We do need to assume that this sort of reasoning is something we have in common, but if we don't assume that we don't even have grounds for believing that we can communicate with each other in any way. So this approach basically protects mathematics from any level of skepticism up to the most debilitating (see Kripke/Wittgenstein/etc.), which is about the best we can hope for.
So where does the "shared understanding" come in? Well, it's actually still crucial: our confidence that we can actually "translate" everything back to the formal realm - to the extent that we are so confident! - relies on an agreement about how this translation process works. Let's look at $\mathbb{N}$ in particular. We have an informal notion of "natural number" that we're trying to formalize, and as a community we've settled on a particular approach which will tell us (partly) how to translate natural language sentences about natural numbers. But one could reasonably argue that that approach doesn't correctly capture one's intuition about the naturals.
Of course, there are a couple objections to this:
Most obviously, this doesn't match up with how we actually think about math while doing it (for most of us anyways). This "default" perspective is indeed based on common intuition: we have a "meta-language" that we're reasoning in, and we tacitly assume some shared assumptions. The formalization approach above can be taken as a safeguard: it ensures us that the work we're doing, regardless of how we conceived it, can be made completely concrete and unobjectionable (we just don't bother doing that).
Even granting that it's valuable to embed everything inside one big formal system - which, FWIW, I agree with - we still have a serious problem: how do we justify which system we've chosen to use as our foundation? There's an obvious "complexity" requirement - whatever formal system we use has to be "Turing-complete" in a precise sense - but that's not a sufficient condition, since we also want our foundation to have some mathematical content itself but it's hard to articulate from a purely formalist stance what that should even mean.
Nonetheless, for someone who's concerned with exactly what we need to make mathematics work, the response above does in my opinion capture this minimal assumption needed: that we have a commonly-understood formal system that we're (implicitly) embedding all our mathematical discourse inside.