Vedanta hinges on the belief that everything, without any exception whatsoever, is the
one spirit. Other orthodox systems, while evolving out of the Vedas, are non-Vedantic,
specifically Samkhya and yoga, among others (see: Patanjali, et al., on the Non-Self).
And of course, there is Buddhism, regarded as both non-orthodox and non-Brahmanic,
(in this context, orthodox means tolerated by Brahmanism).
One way Vedanta approaches the self is to say that anything the self observes can not
be the self, or,"not this, not this" to any feature which can be observed - the body, thoughts, imaginings, feelings, emotions, desires, attitudes, deeply-rooted ideas about the world, and, the idea of oneself as a subject of experience, and as an observer. The principle applied is that the object itself can not be the subject - "I am not this body being observed." "What can be observed is material... but the witness of them all, of all materiality, is of a different order. It is spirit...to know that spirit is not to know an object. It is to realize that one is spirit." (The Essence of Vedanta - Hodgkinson)
In addition, Vedanta teaches that there is a self which is identically the same in each
person, a universal self. The universal self is like space, it has no limits, it is the transcendant Brahman. "Brahman alone is indestructible. As it has no parts, and no attributes, it can not be destroyed by the decay or removal of them. It's nature is existence itself [sat], and existence can not cease to be." For the student, the realization that there is not one iota of difference between the universal self and the individual self is the only way to approach Brahman.
Consequently, Vedanta is more exactly called Advaita (non-dualistic) Vedanta,
because it's elusive truth is that "I am Brahman", or, "All this is verily Brahman". It denies the duality between I and I, I and you, I and he/she, as well as I and the world. However, as a warning to students, lest they think they know of the nature of Brahman, I quote the 8th c. AD philosopher Sankara, perhaps the most authoritative commentator on the Upanishads and Vedanta:
The Absolute [Brahman] is that in which there is no particularity. There is
no name, no form, no action, no distinction, no genus, no quality. It is
through these determinations alone that speech proceeds, and not one of them
belongs to the Absolute. So the latter can not be taught by sentences of the
pattern "This is so and so." In such Upanishadic phrases and words as "The Absolute is Existence-Consciousness-Bliss" the
Absolute is artificially referred to with the help of superimposed name, form,
and action, and spoken of in exactly the way we refer to objects of
perception, as when we say "That white cow with horns is twitching."
But if the desire to express the true nature of the Absolute, void of all external
adjuncts and particularity, then it can not be described by any positive means