A dome is:
1. the best shape for taking stress from very heavy trucks putting all their weight on them without the manhole itself gradually bowing, and
2. is best at transferring that stress equally into the manhole wall (cast concrete cylinder) itself. (A square manhole + manhole cover would disperse force unfairly, potentially gradually cracking the manhole walls / requiring stronger walls. A flat circular manhole would disperse force upon the center of the manhole equally onto the manhole walls, allowing for lower-material-cost manhole walls. A domed manhole cover additionally disperses force from most points on the dome equally into the manhole walls — important, as vehicles won't necessarily be driving over the exact center of the manhole!)
...but really, this is the wrong direction to work in. The original reason manhole covers are round, is simply that the walls of a manhole are best made round, for the same reason drink cans and barrels are best made round: a closed cylinder is great at taking compressive force from a lid above; passing it through as soft, equal tensile force through its walls without buckling strain; and then turning that force back into an equal compressive force on the floor / subsurface.
Most manholes are generally small closed cylinders acting as maintenance areas for nearby pipes, with the pipes coming in through the sides of the manhole walls, and the concrete bottom floor of the manhole resting upon compacted earth.
In this situation, any shape for the manhole other than a cylinder — if driven over for years/decades by cars — would gradually pound the uneven force acting upon the manhole's floor into the earth below, unevenly accelerating soil subsidence. Eventually, you've created a sinkhole below the road, right outside the manhole wall on one side.
The hole came first. They dig a hole, then they have a need to put a cover on it. Making a circular cover to fit over a circular hole is if anything cheaper and easier than making a square cover over an inscribed circular hole, at least when working with metal.
The hole drove the design of the cover, not the other way around.
Refer, after that, to the process of constructing a manhole (https://www.envirodesignproducts.com/blogs/news/how-are-manh...).
At the end of this process, you have a square hole in the pavement, opening to a square excavation, bottoming out at a square concrete foundation, on which has been set a round concrete cylinder, which is then surrounded out to the edge of the square hole with packed earth.
Given this, you could equally-well finish this job either:
1. by placing a square of metal to fill the entire square packed-earth space you've constructed (as when bridging a pothole with a temporary steel surface plate);
2. or by first paving over the exposed packed-earth part, and then placing a circle of metal to cover only the manhole entrance itself.
...which is why people do justifiably ask why, in practice, we seem to always favor option 2 over option 1.
The manhole opens up into a vertical tube, often with a ladder built into the wall, big enough around for a man to descend into a subsurface structure. Hence, manhole.
Manhole covers are always round because the tube they connect to is round, and that tube is round for structural reasons.
A manhole cover has two purposes: people to traverse, but much more importantly, vehicles to traverse.
In the UK manhole covers are generally square.
You'll find square hole covers all over, they're not rare at all. The strength of the lid and its ability to be supported is not at all an issue on a single human sized hole - it's a thick, relatively cheap slab of metal over a small opening. They're the strongest surface of most roads, by a HUGE margin.
https://www.randb-uk.com/product-category/ductile-iron-manho...
[1] This is defined as the minimum perpendicular distance between parallel lines bounding the shape.