Clubs are not supposed to depict actual clubs: that the word has come to be the same is, I think, the result of a linguistic confusion. The suit as it came from France was originally "trefles", or "clovers", hence the distinctive clover pattern and clubs' being green in a four-colour deck. That is still the name in French.

In Italian and Spanish decks however the suit is clubs, represented with what is more obviously a club. Tarot is similar. Latin decks had been more common in England historically but were supplanted by French ones some time in the late Middle Ages. I suspect that the similarity in "clover" and "club" led people to use the latter word as standard in English, even though we retained the French symbols and thus the clover.

Note that we also use the Italian name for spades, although again the symbol was taken from French (pikes): in that case it might have helped that the symbol for spades more obviously resembled a spade. Diamonds seems to have developed independently, presumably from the shape, while hearts still uses the original term as in French.