My understanding is the primary reason is because circular covers cannot fall into the hole itself, whereas that problem exists with other shapes like squares.
Shapes of constant width (e.g. £1 coin) won’t fall into the hole either. Neither would an equilateral triangle.
Oh, and by the way in my country we have square-shaped manhole covers too :O
The OP mentions curves of constant width:
https://en.wikipedia.org/wiki/Curve_of_constant_width
Which includes circles and Reuleaux triangles, which are much more difficult to manufacture than circles. I think this can be rounded off to "circles are the only well-known shape that can't fall into a similarly-sized hole".
Still, as I’ve said, there are square-shaped manhole covers too.