i am not sure about the rigourous definition of isomorphism between graphs but I think it prolly might be smth like two graphs are isomorphic if there is a bijection such that x and y are connected with an edge iff f(x) and f(y) are connected with an edge..
Basically, isomorphism is a mathematician rigorous way to say "if you do this certain twist, certain moving around with this mathematical object, or just label it right, it will basically be the same thing as this other seemingly difference mathematic object"