I get the impression -- see MN 22 -- that statements about self are described as a "view" (diṭṭhi), more particularly a wrong view (diṭṭhigata) ... or a "doctrine" (attavādupādāna -- Ven. Sujato translates that as doctrine but this definition says it means more literally "clinging to the belief in self") ... or "seeing" (samanupassati).
I think there are several kinds of view (self-views, or views about the self) described:
One in MN 22:
They regard form like this: ‘This is mine, I am this, this is my self.’ They also regard feeling … perception … choices … etc.
These are views about the aggregates, taking the aggregates as self, which I think is called sakkāya-diṭṭhi and/or what sakkāya-diṭṭhi is, where sakkāya-diṭṭhi is one of the fetters, and kaya means something like "body".
Others in MN 22, e.g.:
The self and the cosmos are one and the same. After death I will be permanent, everlasting, eternal, imperishable, and will last forever and ever
Others (perhaps more generally) of the style described as examples of wrong view in MN 2:
"There is the case where an uninstructed, run-of-the-mill person... does not discern what ideas are fit for attention, or what ideas are unfit for attention. This being so, he does not attend to ideas fit for attention, and attends instead to ideas unfit for attention... This is how he attends inappropriately:
'Was I in the past? Was I not in the past? What was I in the past? How was I in the past? Having been what, what was I in the past? Shall I be in the future? Shall I not be in the future? What shall I be in the future? How shall I be in the future? Having been what, what shall I be in the future?'
Or else he is inwardly perplexed about the immediate present:
'Am I? Am I not? What am I? How am I? Where has this being come from? Where is it bound?'
"As he attends inappropriately in this way, one of six kinds of view arises in him: The view I have a self arises in him as true & established, or the view I have no self... or the view It is precisely by means of self that I perceive self... or the view It is precisely by means of self that I perceive not-self... or the view It is precisely by means of not-self that I perceive self arises in him as true & established, or else he has a view like this: This very self of mine — the knower that is sensitive here & there to the ripening of good & bad actions — is the self of mine that is constant, everlasting, eternal, not subject to change, and will endure as long as eternity. This is called a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. Bound by a fetter of views, the uninstructed run-of-the-mill person is not freed from birth, aging, & death, from sorrow, lamentation, pain, distress, & despair. He is not freed, I tell you, from suffering & stress.
"The well-instructed disciple of the noble ones... discerns what ideas are fit for attention, and what ideas are unfit for attention. This being so, he does not attend to ideas unfit for attention, and attends [instead] to ideas fit for attention... He attends appropriately, This is stress... This is the origination of stress... This is the cessation of stress... This is the way leading to the cessation of stress. As he attends appropriately in this way, three fetters are abandoned in him: identity-view, doubt, and grasping at precepts & practices.
(e.g. "I exist", or, "I don't exist")
I think they're described as wrong, not because they're true or untrue, not because the self is or isn't an illusion, but because it's a result of attending inappropriately, and not attending appropriately.
And (importantly) it's "wrong" in the sense that it leads to suffering (which I think is like saying that it's unskillful or unwise) -- MN 22 again:
Mendicants, it would make sense to be possessive about something that’s permanent, everlasting, eternal, imperishable, and will last forever and ever. But do you see any such possession?” “No, sir.” “Good, mendicants! I also can’t see any such possession.
It would make sense to grasp at a doctrine of self that didn’t give rise to sorrow, lamentation, pain, sadness, and distress. But do you see any such doctrine of self?” “No, sir.” “Good, mendicants! I also can’t see any such doctrine of self.
It would make sense to rely on a view that didn’t give rise to sorrow, lamentation, pain, sadness, and distress. But do you see any such view to rely on?” “No, sir.” “Good, mendicants! I also can’t see any such view to rely on.
Mendicants, were a self to exist, would there be the thought, ‘Belonging to my self’?”
“Were what belongs to a self to exist, would there be the thought, ‘My self’?” “Yes, sir.”
“But self and what belongs to a self are not acknowledged as a genuine fact. This being so, is not the following a totally foolish teaching: ‘The self and the cosmos are one and the same. After death I will be permanent, everlasting, eternal, imperishable, and will last forever and ever’?”
“What else could it be, sir? It’s a totally foolish teaching.”
“What do you think, mendicants? Is form permanent or impermanent?”
In summary I think the dhamma advises people to not "acknowledge 'self' as a genuine fact".
See also How are 'conceit' and 'identity-view' not the same? which explains that "I am" may arise as a (transient) conceit instead of being a (established) view.
I think that modern "philosophy of science" describes a "theory" as being something which:
- Fits with and explains observed facts (i.e. previous observations)
- Makes useful (usable, beneficial, testable) predictions about future observations
According to that system I'd gather that the Dhamma says that "self" is a type of failed theory, which more or less fails both of these criteria (doesn't fit observed facts, and isn't beneficial).
Is the self an illusion or is it like an illusion?
I think it's like an inappropriate (unskilful or even harmful) theory, for example like "the world is flat" or "wouldn't it be fun to put my hand in the fire".
Incidentally when I was young and before I had access to any reliable literature on Buddhism (but instead only paraphrases by modern authors) I heard it said that "the world is an illusion" and similar statements -- but I didn't find that useful, constructive, prescriptive.
The whole question of whether 'self' is an illusion is still a type of self-view IMO -- "do I exist? do I not exist? is self an illusion? is it like an illusion? etc." -- e.g. as listed in MN 2 above, i.e. part of the "thicket of views."