First of all, the fire in Chicago was historically a small fire, compared with comparable disasters.
1776 fire in New York(usa) 500 death
1871 fire in Chicago(usa) 200-300 death
1881 fire after explosion Wien(austria) 500-700 death
1906 fire in San Francisco 700 death
1943 bombardment on Hamburg(germany) 45 000 death
1945 bombardment on Tokio(japan) 80-100 000 death
I think bombardment is comparable to fire in this context, due an underground city isn't bombardable in that manner.
However, back to topic Building an underground city in 1871
Is this even plausible?
Why not,...
The London tube was finished in 1863, so the ability to build megastructures in the ground was given.
Current has been discovered long time ago and electrical light has been in use since 1840(at least).
Even in ancient Rome they had pumps(powered by people, animals or rivers) to build drydocks, so water in the city would be an solvable Problem.
What impact on history would that have?
Sure,... Question like these are highly speculating but I want to
portray my idea:
In the late 19th centurie, industrialization has reached a peak.
Productivity was the highest value in the part of the world we call the "West" today. Working class riots begun some decades before and there had been a giant change in society.
At this time, there where several concepts (Capitalism, Communism, Socialism) which competed for implementation.
Capitalism requires the ability for rapid economic growth.
This means, the relatively very costly process of building underground would be an problem, it decelerate rapit scaling.
In that manner, capitalism either couldn't stand a chance against the other concepts, or capitalism had prevent underground cities.
What kind of society would we have, if capitalism had not been implemented?
This cannot be answered reasonable.