"Deathless" typically translates amrita (lit. "no-death"), also translated as "the nectar", using the useful parallel with Greek mythology. Another translation of "to attain deathless" is "to partake of the nectar" - just another metaphorical way to refer to The Fruit. I suppose we could compare this with the modern American idiom "drank the Kool-Aid" - although the latter has rather satirical connotations.
More specifically "the deathless aspect" (amrita-dhatu) is (seems to be?) a synonym with what Mahayana calls "the absolute truth" -- the perspective of Three Marks Of Existence, Emptiness, and Pratityasamutpada -- as opposed to "the conventional truth" -- the perspective of Rebirth, Karma, and Samsara vs. Nirvana dichotomy.