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?
This law cannot be false or true, it is just a definition. It already causes problems when applied to non-atomic physical entities over time, such as the ship of Theseus paradox.
In a lot of contexts, this definition is part of arguments and proofs, like many other definitions. If any of the definitions are rejected for the context of the argument, then the argument or proof becomes invalid. The law of identity is not special in that respect.
A similar easier example is logical truth. In classical western logic (attributed to Aristotle), a statement can only be true or false. What if that was false? It is also just a definition, and alternative logic systems exist, such as multi-valued or fuzzy logic. However these systems did not "break" all existing arguments and proofs that had been done with 2-valued logic. Instead, they allowed to apply logic in additional situations where 2-valued logic was limited.
Similarly, there may be alternatives to the law of identity which can extend the ability to reason. In math, "Multiplicative group of integers modulo n" could be seen as such an example in which to different numbers are considered equal to each other.
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
Does philosophy fall apart without the law of identity?
Well, without the Law of Identity, a lot of philosophers have a lot of explaining to do.
Rene Descartes decided, "I think, therefore I am." If it is not true that each thing is the same with itself and different from another, then he cannot say conclusively that he is the thing that is doing the thinking, nor that he is the thing who is.
The list of new problems could continue indefinitely. If A is not identical to A and different from all B, then Bertrand Russell never proved that 1+1 = 2. The sum could be 3, 4.5, or any other number.
Without a Law of Identity, thinkers would have to find workarounds to cure resulting problems.
The laws of identity are unavoidable in philosophy as an assumed context is constant, this assumed context is identity itself. The original laws of identity are contradictory if applied under the Munchauseen Trilemma:
"P" is an assumed variable as a point of view of the observer.
(P=P) leads to an infinite regress as ((((P=P)=(Q=Q))=(R=R))=(S=S))=....
(P=P) has the same premise as the conclusion thus is circular.
Dually each of the laws is subject to the trilemma:
(P=P) is subject to circularity as P is both the premise and conclusion.
(P=/=-P) is subject to infinite regress as -P equates to (R,S,T,...) as variables which are not P
(Pv-P) is subject to assumed assertions as P and -P are strictly taken without proof.
Dually the laws are contradictory if applied to themselves in a circular self referential manner:
((P=P)v(-P=-P)) necessitates under the law of excluded middle one principle of identity exists or the other thus negating the principle of identity into existing in seperate states of either one identity or the other.
(P=P)v(P=/=-P) necessitates that under the law of excluded middle either the law of identity exists or the law of non contradiction. ****If one is false, then P=-P either way. If (P=P) is false then (P=-P) and (P=/=-P) simultaneously. If (P=/=-P) is false then (P=P) and (P=-P) simultaneously
((P=P)=(-P=-P)) necessitates under the law of identity that two opposing values are equal through the law of identity thus negating the law of non contradiction where P cannot equal not P.
((P=P)=/=(-P=-P)) necessitates under the law of non-contradiction that two principles equal through the law of identity are not equal thus the law of identity is not equal to itself.
((P=P)=(-P=-P)) v ((P=P)=/=(-P=-P)) necessitates either the law of identity or the law of non contradiction results, thus negating either the fallacious use of the law of identity or the fallacious use of the law of non-contradiction but not both. Either the law of identity or the law of non contradiction is negated. If the law of non contradiction is negated then the law of identity ceases to exist as P = -P. If the law of identity is negated then the law of non contradiction is negated as P = -P.