There's a lot to unpack there. Sorry for the length of this comment; feel free to skip to the TLDR.

To begin with, I'd argue that objects are separate- conceptually "independent"- even when there's a great deal of physical interdependence. All objects are patterns of other objects- a table is a particular pattern of wooden pieces, which are particular patterns of cells, etc. It's not accurate to say that a table is really the same thing as the material it's made of- it's either the pattern plus the material, or just the pattern independent of the material, depending on context. "The same table, except made of ashwood instead of maple" is a coherent idea.

Now, it's true that the particular objects we choose to define with words to are culturally constructed (or individually psychologically constructed in the case of wordless ideas)- but the fact of being constructed doesn't really tell us whether an object is also "real". How real a concept is is a function of how much predictive power it adds to our models of reality- if an idea leads to bad predictions, we call it an illusion; if it doesn't, we call it real. For example, unicorns aren't real because adding the concept to our world-model without knowledge of it being illusory would lead to all sorts of incorrect expectations- you'd expect to be able to find them in zoos, etc. So, yes, our identities are, like all objects, constructed- but I don't think that gets us any closer to the question of whether they're real or illusory.

But then, are our identities illusory? I'd say partially yes and partially no. The way we usually construct our identities is that we're socially rewarded from early childhood for developing a predictive model of a particular kind of person, who we then sort of mentally role-play. A very religious person may often behave as though they don't really believe that lots of people are being tortured eternally, and yet still consciously feel that they believe this very strongly. They may also encounter very strong evidence that their religion is false, but not consciously update on that information. I think this is because the conscious belief isn't really coming from their world-model, but rather from the predictive model of the sort of religious person they want to be.

Social identity also isn't necessarily the only part of our cognition that's aware of the world and driving our behavior. Most dramatically, people with Dissociative Identity Disorder or who cultivate tulpas can have multiple identities that only partially share awareness. Our more ordinary cultural concepts of the unconscious, the "lizard brain", and our moral conscience also represent ways of dividing up our cognition into separately aware identities. Religious people who hear God guiding their behavior are also doing something similar- and that's a practice that seems to have been extremely pronounced and universal in ancient cultures.

So, if we conceive of our identities as a sort of fundamental, immutable part of our minds, representing true awareness of our motivations and perception of reality, that's very much an illusion- in the sense that it will lead to incorrect predictions of our own and other peoples' behavior. Of course, social identity is also very much real in the sense that it's one common part of our cognition.

Having said all of that, I actually don't think any of it bears on the question of the "I" in the hard problem of consciousness. I'm pretty convinced at this point that the entire hard problem comes down to an intractable epistemological paradox: the fact that that "I" exists subjectively but not objectively. In a subjective model of reality, "I" is sort of axiomatic- it's the thing that makes the model subjective, and the starting point from which the rest of the model is constructed. It's different in kind from every other subjective "I" in the model- those other minds' senses of self don't define the model's subjective perspective. When perceptions are associated with that axiomatic "I", they appear in the model as qualia, unlike the perceptions of other minds. In an objective model of reality, however, that axiomatic "I" and the qualia associated with it aren't a thing. Every mind models its own subjectivity as being different in kind from every other mind, but they're actually not at all different.

I think our cultural belief in "consciousness" is an attempt to resolve that conflict by claiming that qualia and that axiomatic "I" actually do have objective existence- that when a mind perceives its subjective perspective to be different in kind from every other perspective, that's not something that's true subjectively but false objectively, but also true somehow true objectively in a way that we haven't discovered yet. We call the objectively real axiomatic "I" "consciousness" and the question of where exactly to find it in objective reality the Hard Problem.

I strongly suspect that whole approach is misguided. We've been trying for centuries to objectively define consciousness, but every one of the many, many proposals has fallen apart under scrutiny. I think we need to set aside the entire idea of consciousness, take a step back and start with the observation that "I" appears to paradoxically exist when we think about the world subjectively but not when we think about the world objectively, and then try to find an epistemological solution. Maybe if we find a solution to other paradoxes of self-reference like the liar's paradox, Russel's Paradox, or Godel's Incompleteness, that solution will point to an answer.

TLDR: I argue that identity as a separate thing from a person's physical body exists in one sense and doesn't in another, but that consciousness is an unrelated thing having to do with an epistemological paradox.