Why did the industry choose these seemingly unintuitive values of 110V and 220V for the mains AC? Is there a physical reason why they aren't simply 100V and 200V?
They did choose nice round metric values of 100V, 200V and 500V.
And then, they wanted to increase transmission capacity without really adding copper to the system, so they made a series of sub-5% "bumps". (this also made overload-induced voltage drop seem less severe.) The gentle bumps were mild enough that people's light bulbs (the primary load of the day) didn't burn out that much faster, and of course light bulb makers tuned their products quickly with each bump.
By the time Edison was trying to popularize DC, the bumps were up to 110V -- DC, mind you.
When Edison threw in the towel on DC, they chose an AC voltage which would still work with the same DC light bulbs, which worked out to be the same as RMS, so 110 V AC. And this was aggressively marketed to the general public, which is why "110V" stuck in the same way "Xerox copy" stuck.
500V streetcar and subway voltage similarly got several bumps to 600V by that time.
All of them have since had several more bumps. Streetcar and subway voltage has bumped to 750VDC without deleterious effect on motor commutators. BART made a leap to 1000VDC but that was too much for the commutators and they had to begrudgingly back off to 900V.
North American AC power long ago bumped to 115V, some 117.5 and finally 120/240V. There's talk of 125V and everything in the system is insulated for 125V.
Europe did the same thing, UK bumped a little more than the mainland and is now un-bumping to harmonise.
'It's the history of how we got here' is a rather unsatisfactory answer, but it is true, if you mean 'why those exact figures?'
But we can explain why those figures are so close together. There's a rather limited range of usable voltages for domestic power distribution.
Low enough to be safe to touch with bare hands, <50v, would entail massive copper conductors to shift any practical power at all, a kettle would need a 60A outlet at 40v.
When the voltage is lower than 350v or thereabouts, it's relatively safe to handle with casual insulation practices. The voltage will not jump through cracks and pinholes in insulation, any thickness of dry clothing will protect from casual contact, and dry skin will often prevent a fatal shock. So at this level, few enough people die accidentally that people accept the system as 'safe enough', rather like cars! You need to be an expert to handle higher voltage, and the world is not full of experts.
If you read the history of the development of electrical systems around the world, you will find a whole slew of distribution voltages and frequencies. Gradually, the minority choices died out, and today we are mostly left with just the 'big two'.
Why did Thomas Edison plump for 110v? Who knows, but more than 100 probably had a nice ring to it, appealing to both his showmanship and his fledgling market.