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There is no unchanging awareness. When he speaks of "unchanging awareness", Mingyur Rinpoche does not mean that there is an awareness that does not change. He means that all consciousnesses have characteristics in common. The more we understand (Mahayana) Buddhism, the more we are able to understand this and not take things literally.
Generally, when Mingyur Rinpoche says that awareness does not change, he refers to the nature of consciousness. He means that consciousness is clear and knowing by nature. Whether we speak of anger, attachment, mindfulness, joy, an eye consciousness apprehending blue or an ear consciousness apprehending a birdsong, we speak of phenomena that (1) are not form (2) are clear and knowing. Even anger has the nature of clarity and the function of knowing.
Mingyur Rinpoche is a Kagyu and mainly practices Mahamudra, concentrating and analyzing the conventional and the ultimate nature of the mind. Contemplating the analogy of the mind being like the sky is a way to get at the conventional nature of the mind (clear and knowing) that all instances of consciousness have in common. The idea is not to dive in the object of consciousness, but to find that which characterizes all instances of consciousness and to concentrate on that. It is an object that is always there, at reach, just like the breath, because we are always aware of something. If only we could find what is common to anger, desire, pride, an eye consciousness apprehending shape and color, etc. we would find the conventional nature of the mind. If we look for it and know how to find it, we will. On the other hand, if you were to focus only on, say, anger, then it would be different because there is not always manifest anger arising in your continuum.