In many, many different areas of math, we define abstract objects and structure-preserving maps between them, and then come across a suitable notion of "isomorphism." The idea is always that isomorphic objects share every property that a mathematician in that field would care about: Homeomorphic topological spaces have all their topological properties in common, group isomorphisms have all their group-theoretic properties in common, and so on.
But at the end of the day, we still have to verify by hand that any given property is preserved by an isomorphism in the category we're working in. For example, say we're working in group theory. It is intuitively obvious that under an isomorphism $\phi : G \to H$, corresponding elements have the same order, compose in the same way, $\phi$ takes subgroups to subgroups, normal subgroups to normal subgroups, centers to centers, and any group constructed out of $G$ (direct/semidirect products, quotients, etc) should yield an isomorphic result when $G$ is replaced by $H$. (These are just a few of many examples.) It feels like it should be the case that all of these properties correspond between $G$ and $H$, but nonetheless we have to verify each of them one at a time. The proofs are never hard, and they suggest that there must be a more general way to think about them–some sort of big theorem that says that all "group-theoretic properties" can be transferred from one group to another by an isomorphism.
How could we make this idea rigorous? How would we even define a "group-theoretic property", or analogously a "topological property" or a "linear-algebraic property" or a "manifold property"? I would think the definition would stem from the idea that such properties are those that are phrased using only the structure of a group, (or a topological space, or a vector space, or a manifold); but this still seems imprecise. Assuming we could make this notion precise, could we then proceed to prove a general theorem that all such properties/objects are preserved by isomorphisms in the category we're working in, and then we don't have to tediously prove, for example, that group isomorphisms carry centers to centers, or homeomorphic spaces have the same number of conencted components, etc., because these would all fall out as special cases?
Or is this a futile task? Maybe it so happens that there are weird examples of properties that seem like they should be preserved by isomorphisms but aren't, even though they are phrased using only the structure of the category. Math is full of pathologies, and at this point I can't seem to trust 100% that isomorphisms are these magic structure-preserving identifications that they are always made out to be.
Best Answer
Mathematical logic (specifically model theory) provides a partial answer. Let $M$ and $N$ be structures for a first-order language $L$. $M$ and $N$ are elementarily equivalent if every closed formula satisfied by one is satisfied by the other. $M$ and $N$ are isomorphic if there is a 1-1 map between $M$ and $N$ that preserves all the relations and functions mentioned in the signature of $L$. Theorem: if $M$ and $N$ are isomorphic, then they're elementarily equivalent. See, say Marker Model Theory: An introduction, §1.1, or Hodges A Shorter Model Theory, §1.2.
I think this serves as a reasonable candidate for "a general theorem that all such properties/objects are preserved by isomorphisms in the category we're working in".
I say a partial answer, because choosing the language in each case remains an issue. Let me elaborate for your example of groups. We want to show that being a subgroup, or a normal subgroup, or the center, is preserved by isomorphisms, all in one shot. For $L$, we include the following in its signature: the constant symbol 1, the function symbols $\cdot,{}^{-1}$, and a unary relation symbol $S$ for the subset under discussion. (There are other signatures that would also serve.) Here are the closed formulas that express "$S$ is a subgroup", etc. I'm going to be a bit sloppy for increased readability, using juxtaposition for the operation and omitting parentheses. Also, when I write "$S$ is a subgroup" in the second two bullets, just imagine the first bullet being repeated in full.
So if $M$ and $N$ are isomorphic, then $M$ satisfies one of these formulas if and only if $N$ does—that's what elementary equivalence says. And if $M$ and $N$ are isomorphic groups, then the subsets defined by the relation symbol $S$ correspond, and therefore one is a subgroup (or normal, or the center, or anything expressible by a closed formula in this language) if and only if the other is.
If you're familiar with first-order logic, you'll be aware of various hurdles to overcome. For example, to define "commutator subgroup" with a closed formula, you'd need to expand the language to allow for sequences of arbitrary finite length, since the commutator subgroup is generated by the commutators. That means incorporating $\mathbb{N}$ into the structure in some manner. I don't mean that $\mathbb{N}$ would be a subset of the group, rather that the structure would be (implicitly) an ordered tuple $(G,\mathbb{N},\ldots)$. For "derived series" you'd need to expand the language some more. But all these obstacles can be mastered with standard techniques.
A fuller answer would discuss connecting the category theory with the model theory. I plead limitations of both space and my expertise.