You are welcome.
Let me try to give you something more immediate to your question.
When we begin our studies of "mathematical logic" we have a naive appreciation for the nature of truth that is conflated with our personal beliefs. We talk of "truth" and "falsity", we represent "truth" and "falsity", we formulate a logical algebra with which we investigate proof structure, and we find that we must differentiate between material consequence and logical consequence precisely because of our interest in a logical algebra to be used in a logical calculus.
Where have we actually discussed what we mean by truth?
There are a multiplicity of theories of truth. The two which are most relevant here are the correspondence theory and the coherence theory. A majority of philosophers will discount the coherence theory because it can only have a relationship with material reality by sheer luck.
There are also conceptions of truth. Tarski proposed the semantic conception of truth. The analytic conception of truth probably originates with Kant. It holds that the truth of a statement depends upon the meaning of its words. There is also an epistemic conception of truth. This is closely related to the typical mathematical practice of proving that a definition is non-vacuous. When it involves singular terms, it also involves a uniqueness proof.
You are questioning how one surmises existence of an object by merely using a singular term purporting to denote it. By virtue of its quantifier rules, first-order logic is a logic with existential import. If you accept the logic, you are bound to its presupposition. Free logics are different in this regard, and, Frege's use of the indiscernibility of non-existents had been intended as a well-formedness condition providing for the substantive denotation of his terms.
Now, all instances of reflexive identity are true in an analytic conception of truth. It is assumed that the two occurrences of syntactically identical symbols have uniform denotation (if they denote at all). This is why formalists find the possibility of a false reflexive identity statement confusing. Yet, a semantic conception of truth insists upon the substantive denotation of singular terms in true statements.
Tyler Burge, for one, has held that Tarski's semantic conception of truth clearly justifies the idea of false reflexive identity statements under a correspondence theory of truth. Within mathematical contexts, examine the transitivity axiom from Tarski's cylindric algebras. It has the effect of binding identity statements with the existential quantifier. In particular, reflexive identity statements only hold for existents.
Philosophers and symbolic logicians will have a problem with Tarski's axiom from cylindric algebra. It would, in effect, be an existence predicate and philosophers are taught to reject such things. Symbolic logicians will object to the fact that it has an infinite expansion relative to self substitutions into the definiens. But, again, in mathematical contexts, this seems closely related to the model-theoretic investigation of identity in the presence of apartness done by Statman and van Dalen and the proof-theoretic investigation of the same by Borretti and Negri.
Writing before the more modern developments of free logics and cylindric algebras, Abraham Robinson wrote the paper, "On Constrained Denotation." Its principal motivation had been the description of a language given through the formulation of definite descriptions. One simply cannot use a model-theoretic identity relation because the identity relation is essentially constructed as terms are introduced. From this, one may surmise that an epistemic conception of truth is a constructive methodology. Using ideas from free logic makes the epistemic conception more understandable. But the first-order logician who insists upon an extra-logical stipulation of the domain will find it unpalatable.
Now we can compare the correspondence theory and the coherence theory.
Our naive intuition is correspondence. In a sealed room with one chair we say "the chair", using the definite article referentially. In a restaurant we might invoke the ostensive language act of pointing to a table while uttering "that table." But if we apply a correspondence theory of truth to mathematical objects, we seem committed to the existence of mathematical objects without any ability to produce such an existent. To a large extent, we reach this point because of commonly held beliefs concerning a relationship between mathematics and material reality.
This is why I began by pointing out how we begin our studies with the conflation of truth with belief.
Kant had probably been the first to propose a coherence theory of truth, although there had been no vocabulary to express it as such at the time. He had not been trying to delineate "mathematics". He had been responding to Hume's modern account of skepticism. The argument applies to any correspondence theory of truth intended to make claims about material reality. By relegating mathematics to "intuition" Kant had been trying to provide a basis for "objective knowledge" in contrast with claims of "absolute knowledge" (correspondence with material reality).
Not surprisingly, philosophers -- especially analytic philosophers -- took issue with this. Frege's logical studies arise in this context. Hence one sees the motivation behind his emphasis on truth and his emphasis on extensionality.
Ultimately, these are waters you must navigate for yourself. I find the argument that mathematical objects exist because science is useful to be analogous with the argument that God exists because prayer has efficacy. To be skeptical of the latter without being skeptical of the former is incoherent in my view.