Why did almost every old city wall have a moat around it?

Answer: To stop tunneling and slow siege towers

To stop tunneling and slow siege towers ✓ — Correct! A moat solves three problems at once: attackers can't dig tunnels under the wall (they'd hit water and collapse), siege towers and battering rams can't be wheeled up to the wall, and scaling ladders need to be longer to clear the moat too. It's the cheapest force multiplier in fortification.

To provide drinking water during a siege — Wrong. Moat water was usually stagnant and contaminated — useless as drinking water during a siege. Cisterns inside the wall handled drinking water. The moat's job was anti-engineering.

To raise fish for food in peacetime — Wrong. Some moats did contain fish, but that's incidental. The military function is what made them universal — they made tunneling, ramming, and tower-rolling all much harder.