The English name of this suit is a borrowing from a different pack, and the shape on the card is a third thing again.
Pique, the pike
In French the suit is pique, the head of a pike or spear, and the sign is a pointed leaf-like shape on a short stem. That is what the French designers of the later fifteenth century drew.
Spade, from swords
English calls it spades, from the Italian spade, meaning swords, the suit it displaced in the Latin-suited packs used earlier. The digging implement the modern English word denotes has nothing to do with it. Clubs acquired its English name by the same route, from Italian bastoni. In both cases the old Latin suit name stayed with the slot when the French sign moved into it.
Why the French signs exist at all
The French pattern replaced pictorial suits, German hearts, bells, acorns and leaves and Latin cups, coins, swords and batons, with four flat two-colour silhouettes that a stencil could print. Production cost is the reason this pattern spread and the others did not.
In cartomancy
Readers map the sign onto the sword suit of the Latin packs, which is the source of its association with conflict, severance and hard news. The mapping is a reading convention, and in this case it is also the historical name reasserting itself: the suit is called swords in English by accident and read as swords on purpose.