What would happen to all of philosophical thinking if the law of identity, i.e. 'each thing is the same with itself and different from another', were false?
We have no indication that it is true, as anything more than a linguistic convention that simplifies our expressions.
In terms of physics, it is simply not true that a chair really is just itself and different from everything else. It is a collection of atoms, and which ones are really part of the chair is pretty ambiguous. Beyond that, whether the atoms actually are where they seem or 'tend' to be is even more ambiguous. And the spatial relation that collects them up into a chair is somewhat arbitrary and varies from person to person: Does it include any attached cushions. Is any covering of dog hair part of the chair, nor not?
Humans just like equivalence relations. They limit the number of variations we need to accommodate on a regular basis. If things can be swept into an equivalence class, their details can be ignored.
But anyone who has studied the models we use as foundations for mathematics will realize that equivalence must be constructed, and is not inherent. A real number is not just itself, it is constructed as an equivalence class of something much more arbitrary and formless, and we do not even really agree upon what that is. It can be the set of convergent sequences that eventually converge together, it can be the set of rational numbers that are all less than the same things, it can be all the markable distances that can be exactly overlayed by a non-scaling geometric motion...
So if it is just not true, changing it would not affect much.
As to how we are to survive without it, consider this: https://philosophy.stackexchange.com/a/33731/9166