A myth about a drink nobody can obtain would not by itself organise anything. Amrita organises a great deal, because of a detail attached to the struggle over the vessel: during the flight and the fighting, drops fell to earth. Four places in northern India claim them, and the Kumbh Mela is held in rotation at those four river sites on that account. It is among the largest gatherings of people anywhere in the world, and its calendar is set astrologically, so a story about spilled liquid governs both where and when tens of millions of people travel.
The word travels further than the myth. Amrita is a-mrta, not-dead, a negated participle, and the same Indo-European formation stands behind Greek a-mbrotos and so behind ambrosia. The two traditions inherited one word for the substance that stops death, and each built a different story around it. In Sanskrit the adjective stayed productive: amrita can qualify anything deathless, and the noun is used for the sweetness of a teaching or the taste of liberation, so a term from a churning myth becomes ordinary religious vocabulary.
Buddhism took it up in that abstract sense, where amrita most often means the deathless state itself rather than a liquid, and Vajrayana practice uses consecrated substances under the name. Sikhism gives the word a concrete ritual life again: the amrit of the Khalsa initiation is sugar water stirred with a double edged sword while scripture is recited, and it is a distinct thing from the Puranic drink, sharing the name and the claim about deathlessness rather than the myth. The atlas holds that Sikh substance as its own entry.