I remember reading this question - and it's answer - in a maths puzzle book in my mid teens. It's a very simple solution - and very easy to start investigating further. The short answer: manhole lids are circular so that they don't fall down the hole (risking losing the lid, and landing on a worker who is in the hole). Technically, the lid has a constant maximum diameter irrespective of which angle you use to measure it.

The same cannot be said of most other polygons - let's take some quick examples.

Squares: the sides are shorter than the diagonals, so a small rotation will enable the lid to fall down the hole.

Pentagons: the ratio of side to diameter is smaller, but it's still possible to drop the lid down the hole.

Pentagons: the ratio of side to diameter is smaller, but it's still possible to drop the lid down the hole.

Equilateral triangles are an exception; and in fact you do sometimes see manhole lids that are equilateral triangles (sometimes hinged along one side).

The same principle applies to coins. In order to function correctly, a vending machine has to be able to identify and distinguish different coins, based on their diameter and irrespective of how they fall through the slot. The coins which are not circular are based on Reuleaux polygons, such as the Reuleaux triangle, where the shape has a constant diameter - the key requirement for coins, and manhole covers!