Origins
Every rivalry is, at its root, a fight over water. The word rival comes from Latin rīvālis, meaning 'one who uses the same stream', from rīvus, 'a brook' or 'stream'. Two farmers drawing from the same river were rīvālēs — and the competition that followed was among the oldest forms of human conflict.
Roman law devoted considerable attention to disputes between rīvālēs. Who could draw water, when, and how much? The legal term carried no inherent hostility — a rīvālis was simply someone with a shared claim. But shared claims breed disagreement, and by the time the word reached French and English, the adversarial sense had won out.
Latin Roots
The water root runs through English in unexpected places. River comes from Latin rīpārius ('of the riverbank'), a close relative. Derive meant 'to draw water away from a stream' — to channel it in a new direction. Arrive originally meant 'to come to shore' — ad rīpam, 'to the riverbank'. A rivulet is a little stream.
Shakespeare used rival in its modern competitive sense, and by the 17th century the old 'neighbour' meaning had vanished. But the etymology survives as a reminder that competition begins with shared resources — and the most fundamental resource of all was flowing water.