As something of an aspiring mathematician, I have to say that part of the popularity of isomorphism theorems is that they are very exciting. In the more abstract branches of
mathematics, part of the motivation driving new work is
the goal of seeking out new common structure underlying
previously disparate ideas: if mathematics is the universal language, then let it be as elegant as possible. Category theory and universal algebra were both born out of this line of research. And isomorphism theorems are a precise demonstration of this commonality.
Isomorphisms simplify mathematics.
Within category theory, isomorphisms are also key to many universality properties: often one needs to show that some object is unique. A common trick is to consider uniqueness upto isomorphism instead. After all, if an isomorphism preserves all the structure you care about, why make unnecessary distinctions.