Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?
Any manhole cover bounded by a curve of constant width will not fall through. The circle is the simplest such curve.
A manhole cover can't fall into the hole if the minimum width of the cover is greater than the maximum width of the hole.
For example, consider a one-meter square cover over a square hole slightly smaller than $1\over\sqrt 2$ meter on a side. The diagonal of the hole is slightly less than 1 meter, so the cover won't fit into it.
The point is that manhole covers aren't the same size as the manholes they cover; they have flanged edges.
EDIT :
Oops, I missed this sentence in the question:
This figure has the same shape as the hole in the plane, but infinitesimal larger.
so my answer, though it does have real-world applications, doesn't really answer the question as stated.
This question was frequently asked on technical interviews for software engineering positions, up until developers started using counterfactual reasoning.
There is an excellent article "If Richard Feynman applied for a job at Microsoft" showing that there is actually very little practical link between manhole shape and it's conventional representation as a circle.
If I may, I would like to throw a few quotes:
Interviewer: Why are manhole covers round?
Feynman: They're not. Some manhole covers are square. It's true that there are SOME round ones, but I've seen square ones, and rectangular ones.
Interviewer: I mean, why are there round ones at all? Is there some particular value to having round ones?
Feynman: Yes. Round covers are used when the hole they are covering up is also round. It's simplest to cover a round hole with a round cover.
Interviewer: Do you believe there is a safety issue? I mean, couldn't square covers fall into the hole and hurt someone?
Feynman: Not likely. Square covers are sometimes used on prefabricated vaults where the access passage is also square. The cover is larger than the passage, and sits on a ledge that supports it along the entire perimeter. The covers are usually made of solid metal and are very heavy. Let's assume a two-foot square opening and a ledge width of 1-1/2 inches. In order to get it to fall in, you would have to lift one side of the cover, then rotate it 30 degrees so that the cover would clear the ledge, and then tilt the cover up nearly 45 degrees from horizontal before the center of gravity would shift enough for it to fall in. Yes, it's possible, but very unlikely. The people authorized to open manhole covers could easily be trained to do it safely. Applying common engineering sense, the shape of a manhole cover is entirely determined by the shape of the opening it is intended to cover.