Link's Awakening was set within the dream of the Wind Fish, with Koholint Island and all its inhabitants existing only within that dream. If you perfect clear Link's Awakening DX (by completing it without dying) Marin is implied to have been reincarnated in the "real" world as a seagull. Unawakening is the dream of Marin-as-a-seagull, trying to recreate her memory of her home*. On the player's arrival Marin foresees how they will help the people of the island and end her dream, just as (as Link) they originally awoke the Wind Fish, so she attempts to prevent completion of the trading quest by tearing the recipe page from the library book and throwing it in the bottle into the sea. Of course this washes back up on the beach and the player is able to complete the trading quest regardless. Accepting the inevitability of waking and the impossibility of reliving the past, Marin decides to wake up and the game ends (possibly fun detail - if you have the console open in your web browser for the ending you will see the game code ends by throwing the error "Error: The dream has ended").
The game is called "Unawakening" as an obvious reference to Link's Awakening and this being a tribute that recreates much from the original game without fully replicating it, but it is also a reference to the ending. The final owl dialogue says "The morning draws near... The seagull unawakens." The seagull is, of course, Marin, although the exact identity of the dreamer has been an unfolding mystery throughout the game (much as the island
being a dream is unveiled throughout Link's Awakening). While one usually just
awakens from a dream, the fact that Marin originally existed only in a dream means that for her, her form within the dream is her real one as opposed to her existence as a seagull in the "real" world. Hence she does not awaken, but "unawakens".
*Another of the owl's dialogues references this when he says "The dream is nought but a reflection... The dreamer but a shadow of a dream." and this is meant to be evidenced by the reduced screen size, sprites and palette. The dream of a seagull should naturally be less well realised than the dream of the deity-like Wind Fish. Marin is also something of an analogue to myself as the developer and gives me the perfect narrative excuse for why my game is worse than the original it is based on - I can claim it to be intentional! (While of course we all know it was inevitable
)