compete

Compete comes from Middle French competer, from Latin competere, meaning "to coincide, to be equal to, to be capable of." That Latin verb combines com- ("with") and petō ("to seek, to aim for, strive for").

The facts given don't trace a shift in meaning beyond the Latin sense of coinciding or being capable together, so the honest story is simply that the root idea — striving alongside another for the same aim — carried through into English use.

Latin also gives us competītor, the direct ancestor of "competitor," built from the same competere root.