Not sure if this helps with your confusion/skepticism, but natural language is full of under-determined sentences, e.g.
Today is Tuesday.
That statement is true if where you're at right now (time and space) is a Tuesday and false otherwise. Words like 'today' are known as indexicals in linguistics and the philosophy of language. Think of that formalized as a variable statement that says "x is Tuesday"; it's true for some values of x and false for others. Further, if you lived in world in which no day of the week is called "Tuesday" it would always be false.
In first-order logic, the point of fixing a domain of interpretation for a first order language is precisely so that you can say for every sentence if it's true or not.
Statements like "All S are P" (i.e. more formally, "For all x, if x is [in] S, then x is [in] P") are basically true in some models and false in others. To be able to assign such statements a truth value requires providing an interpretation of the (non-logical constants symbols S and P and of some interpretation of the quantification in terms of what domain it ranges over. E.g. if the domain is the "real world" and the interpretation of S is "x is a man" and that of P is "x is mortal" then the sentence is true in that model. But if the model is (a fictional) one in which all men are immortal, then the same sentence is false for the same S and P.
When someone says (in natural language)
they mean a whole lot of things; that sentence has a lot implicature. They don't want to be understood as stating for example that a dog named God exists! In an unrestricted quantification ∃g can well be satisfied by any dog (in the real world, assuming that's the domain of discourse) or anything else in the domain of discourse. Normally when someone says "God exists" they mean that entity has a lot of properties (being a dog usually not included), i.e. they mean g is omnipotent, omniscient, it has created the world and what not:
the sentence ‘God exists’ does not carry its meaning with it,
independent of context and background beliefs (Phillips 2004, 67). For this reason, implicaturism argues that what varies with one’s religious beliefs and practices is what language additionally conveys.
- Phillips, D. Z. Religion and Friendly Fire: Examining Assumptions in Contemporary Philosophy of Religion. Aldershot: Ashgate, 2004.
In the following expression, why would we need to specify any domain?
¬∀y(∃x(y ∈ ℕ ∧ x ∈ ℕ ∧ y = x²)
You're confusing two (other) things here: the domain in which interpretation is provided for a (first order) sentence and syntactic use of ∈ in a first-order language that includes the (ZF[C]) axioms of set theory! ZFC's (nine) axioms basically define (syntactically) the properties of that sole binary predicate denoted by ∈
ZFC is an axiom system formulated in first-order logic with equality and with only one binary relation symbol ∈ for membership. [Axioms omitted]
If you want an analogy here, if you want "God exists", as a first-order language formula, to (syntactically) mean more than the mere triviality of some element existing in the domain of discourse/interpretation, then you then need to specify what axioms God fulfills!
As for that "math expression" you gave, y ∈ ℕ in that expression is surely meant for it to be interpreted in the context of set theory, i.e. considered as added to axioms if you want to prove something follows from that logic formula in ZF. But you also need to define what x² and ℕ means, which the axioms of ZF alone don't do for you. Note also that you can define natural numbers (albeit not completely) via Peano's axioms, which don't actually need to include ZF, and in that context there's no need to write ∈ or ℕ.
[the successor,] the addition[,] and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.
And the PA axioms (which have 3 predicates) look like
The don't carry any ∈ in them (which isn't among the 3 PA predicates), and you'll note that the quantification is thus not "restricted" to anything.
But normally a "math statement" is by default interpreted in the language of ZF[C], plus whatever might be "well understood from context", e.g. what ℕ means, unless otherwise specified.
If you don't want to take my word on that, here's a mathematician saying it
the theory [i.e. axioms and whatever follows from them] ZFC specifies that there is one relation symbol ∈, which we usually read as "element of". A model of ZFC interprets the symbol ∈ as a binary relation, so in this sense it defines its meaning.
[...] most mathematicians seem to be a bit unclear about this distinction, as they never really have to distinguish the language they use from its [intended] meaning.
-- Andrej Bauer
And another concurs (you many want to read all their comments as they are insightful by giving some more examples of this practice, including arithmetic, but I'm not going to copy all of that here).
It is very common, when talking about a first-order theory, to conflate the symbols in the theory with their intended interpretations. [...]
even though the elements of an arbitrary model of ZFC might not "really" be sets, or the interpretation of the ∈ symbol may not really be set membership, we often speak as if they are. The key observation is that, if someone "lived inside" the model, and only had access to the ∈ relation, that person would have no way to tell that the things they see are not sets. One way of making this observation precise is the following [Mostowski Collapsing] lemma, which is proved from "outside" a model of set theory. [...] This lemma says that if we look from the outside at a model that looks even vaguely like a (well-founded) model of ZFC, we can replace it with an isomorphic model whose elements are actually sets and whose binary relation is actually set membership.
-- Carl Mummert
So yeah, you're confusing syntax and semantics here, but you may excused for doing so. That "math formula" you gave is surely meant to have an interpretation in a set universe though, but it's not what it intrinsically carries with it, except by social/math convention(s) and possibly the [con]text given before it in the work where it is found. (You could, e.g. define/intend '∈ ℕ' to mean [be interpreted as] "is beautiful", which of course nobody could guess unless you were explicit about that.)
(There's also that technical sense-- Mostowski Collapsing lemma --in which it is somewhat appropriate to forget about the distinction between language and intended model of ZFC in a lot of "run of the mill" math.)
When a mathematician writes ℕ you may think you know what it means because you were probably taught that symbol with that meaning in school. But that doesn't mean the first-order sentence you've shown carries that meaning for ℕ... except by implicature! This is almost the same as when someone says something about God actually, except that in the ℕ case there's less ambiguity about what properties we expect from it. (There surely are less obvious terms in math, e.g. when a mathematician speaks of a ring or a group, he [usually] doesn't mean it in the sense of common day usage of those words. And there are also such terms that have multiple definitions in math itself, e.g. "filter".)