Firstly, what we mean by “system” is what's called Pramāṇa – or “means of Knowledge”. In Advaita Vedanta case, it's a means of Self-Knowledge.
Means of knowledge means an instrument, using which, one arrives to the direct understanding themselves. Just like mathematics, is a sophisticated means of knowledge for arithmetic.
Like mathematics, a pramana consists of concrete concepts, something your mind can work with long enough, until the self-knowledge converts into spontaneous and effortless vision of Truth.
Self-Knowledge in stage of “listening” can be equated to object knowledge – as in “I am now learning about the Self, and that I am that Self, even though I can't see how that's possible“.
Just like when first learning to drive. The road rules and codes are object knowledge, which you are learning – as if it's something “out there – at some time“.
But once you're in the field driving, over some time, these rules/codes get integrated as “here NOW” – while driving.
Similarly, through practice of Vedānta pramāṇa, the definition of “Self-Knowledge” gets converted from object knowledge to “here NOW, not away-from, not separate-from, instant” Knowledge of the Self – AS the Self (Ātman).
Thus Vedānta is pramāṇa for Self-Knowledge as revealed by ṛṣis (sages) in deep meditative contemplations. And this revealed Knowledge does NOT contradict your own experience, thus it's impossible to refute or disprove (else it would've been done by now throughout hundreds of years of counter philosophies; in which case, Upaniṣads, Bhagavad Gītā and Brahma Sūtras wouldn't have survived).
Furthermore, the revealed Knowledge is verifiable through logic. Vedānta prāmaṇa is not based on beliefs which can't be verified through your own experience (assuming mind is purified enough to see).
Let's find out more about Vedānta prāmaṇa – compared to other “systems”…
- Vedanta differs from other teachings or religions, in that it is a comprehensive viewpoint of multiple ṛṣis who saw the truth. As Chinamayananda use to say: “Vedantic teachings are inserted into the fabric of creation itself and so the rishis were able to access them in deep states of meditation.” All other teachings/religions, to the best of my knowledge are based to a large extent on the experiences of their founders. There is no founder of Vedānta and so we don't have to go off an individual's experience like many other teachings do.
- Vedānta (as it should be taught) is a highly systematic and scientific revelation of Self-Knowledge. If one listens to a qualified teacher, then initially (in the Śravanam phase) an open mind and faith in the teacher is required. After that, (in the mamanam phase), the teacher actually encourages the student to express all their doubts which the teacher addresses. In most other teachings, faith is encouraged but doubts are not. To me, this is an important point which separates it from other teachings: Vedānta has the quiet confidence that it can answer all doubts of the student.
- Vedanta offers a complete cosmology. It shows you that you are Consciousness. But it also describes Maya in detail. It describes how Māyā manifests (guṇas, three bodies, 5 sheaths/pañca kośa, etc). It describes the relationship between Maya and Brahman. To the best of my knowledge, no other teaching or religion is able to describe creation in a way that actually aligns with modern scientific thinking (eg: in physics and neuroscience).
- Finally, the practices offered by the Vedas and Vedanta for purifying the mind (Karma/dharma yoga, upāsanā yoga, bhakti) are so relevant to today's lifestyle. That is, Vedānta does not discount the Jīva (the ego self with a personality, a life-story). Through these practices, it makes life easier in a practical and useful way for the Jīva.
Extra reference: Buddhism, Yoga and Vedanta