The four suits of the tarot came to Europe on ordinary playing cards before any pack carried trumps, and their history is the history of card playing rather than of divination.
An unchanged suit
A surviving Mamluk pack held in the Topkapi Palace collection in Istanbul carries four suits: cups, coins, swords and polo-sticks. Cups passed into the European Latin suits unchanged, as coins did. Both are objects that needed no reinterpretation by the copyists who brought the pattern west.
The suit is coppe in Italian packs, copas in Spanish ones, and coupes in the French Latin-suited packs, all of them the plain word for a drinking vessel.
What became of it in the French suits
When the French suit signs were devised in the later fifteenth century, they were simplified shapes that a stencil could print. The cup became the heart, and the descent is visible in the correspondence tables that cartomancers have used ever since: hearts to cups, tiles to coins, clovers to batons, pikes to swords.
Water
The assignment of this suit to the element Water is one of the more stable parts of the occult correspondence scheme, agreed across systems that disagree sharply about Wands and Swords. Stability is not antiquity, though. The elemental attributions were attached to the suits from the eighteenth century onward and formed no part of the pack as it was played with for the first four hundred years of its European life.
Sources The Game of Tarot: From Ferrara to Salt Lake CityMichael Dummett with A Wicked Pack of Cards: The Origins of the Occult TarotRonald Decker, Thierry Depaulis and Michael Dummett