Four equal sides at right angles. Also a public place a town gathers in — both are shapes made by agreement.
Vulgar Latin *exquadrare (to make square), from ex- + quadrare, from quadra (a square figure); the town square took the geometric name because it is a shape bounded on four sides, not because it is where people gather.
Four equal sides holding at right angles, and the open ground a town keeps for gathering. A shape and a place, both trusted because the angles agree.