The Self is also an illusion though, in all likelihood
An illusion for whom?!
That's an illustration of how deeply ingrained, how compelling, the illusion is. It is scarcely possible to talk about such things without language that assumes the illusion.
So...once again. An illusion for whom?
And once again, the phrasing of the question presupposes there is a 'person' to be illuded. Which is of course entirely reasonable in daily life. It is near impossible to describe what is meant by 'self' with any sort of clarity or precision and in a way that suits all contexts. If I take, as a working definition, a sense of self to mean a sense of inner singularity, personal continuity, agency and perception, then I would say that there must be an illusory aspect to this in the same way there is an illusory aspect to all conscious experience, in that it is all cerebral fabrication of some form that we implicitly take for real. Thus for example I am looking at my screen right now and I accept unquestioningly that my vision of the screen actually is the screen I am looking at. But of course it is not, my sense of sight is a neurological phenomenon happening at the back of my skull but the experience is so sublimely compelling that I trust that my sight of the screen actually is the screen itself. And suppose I reach out to touch the screen to confirm what my eyesight tells me, my sense of touch is another component of the overall immersive sensory illusion of conscious perception. It feels like my finger is touching the screen, but actually the sensation of touch is something fabricated in brain and cunningly back-projected through some sort of reverse proprioception to make it feels as if it is the finger which is experiencing touching.
The sense of self in particular is a component of the overall fabrication that comes as a package under the umbrella of conscious experience, and to my mind there are telling pointers that illustrate the work that goes on behind the scenes to produce this feeling; for example we often look to disorders as a way to illuminate the nature of 'normality' and we find there are schizophrenics that sometimes lose their sense of self, and for instance, when they move an arm, they have the compelling belief that someone else is moving it, not them. This is a case where the sense of self is compromised or underfunctioning through pathology. Another pointer to the divided inner self is the fact that we experience cognitive dissonance, born of the defacto observation that all mammals in essence have two brains that are joined together by a tiny amount of connective tissue, and in human psychology, this often manifests as a tension between two ways of thinking - one intuitive, directly experiential way, and on the other hand, an abstract, logical way of thinking, which might very crudely be portrayed as underlying the tension in human society between religion and science. Pointers such as these suggest to me at least that much of the work that the brain does in generating conscious experience is to do with the homogenising and calibrating and harmoniously integrating disparate and often warring underlying components into a seemingly single point of focus, perception and volition, and this taken together with the persistence of individual memory over time thanks to faithful cell replication, we have an enormously powerful and compelling sense of self, of individuality, of personhood.