Isomorphism implies that there is a complete reversible mapping (a bijection) between structure A and B .

Real number under addition are isomorphic¹ to positive real number under multiplication. Yet they are not the same objects.

1- The mapping is f(x)=e^x. See https://math.stackexchange.com/questions/573794/prove-that-m... for the proof

When it comes to group theory, they are the same object. When we talk about things like rotation groups, we're not usually concerned with the way they are represented (much as we don't usually how the real or the complex numbers are constructed, for both of which there exist multiple different constructions).