The Atman is already enlightened. It is the jiva that is unenlightened, as the jiva is ignorant of his/her true nature.
When an unenlightened jiva sees an enlightened person (jivanmukta), s/he says, "Look, that person is enlightened." They are perceiving them as such because the jiva is perceiving from vyavaharika. Jivanmukta don't perceive themselves as enlightened, as their perception is already in Paramartika. They are already experiencing their true nature in Absolute Reality (Brahman).
Think of this in terms of non-lucid dreaming versus lucid dreaming. If the dreamer is ignorant of the dream, their perception is only within the dream and that dream is their only reality. However, if one is experiencing a lucid dream in which they realize the dreamer is illusory and not their true nature, they have an entirely different perception of the dream itself, as they realize the waking state is the true reality and that the dream state is illusory.
And to answer the question in the title, yes. All Atmans are the same Atman, which is Brahman, though they appear as different Atmans. All waves in the ocean are the same ocean, only each wave appears as an individual wave.