Every class of finite structures closed under isomorphism can in principle be classified by a (countable) list of representatives of the isomorphism types.

I agree that such a list would be uninteresting if it was not computable. I wouldn't insist
on "decidable in polynomial time" as Richard Borcherds did above. Even a list that is computably enumerable (r.e.) might be
interesting for pure math purposes: At least their is a way to effectively (but maybe not efficiently) generate a list of representatives.

There is another approach to classification taken in descriptive set theory:

Consider a class of countable models of a certain theory or of separable objects such as manifolds and so on.

Then typically there is a natural separable complete metric space (a Polish space) of representatives (wrt to isomorphism) of the objects in your class.

Note that I do not require that each isomorphism class is represented only once.

You can usually not get this.

Isomorphism is now an equivalence relation on a Polish space, usually relatively easily definable.

There is a wellstudied hierarchy of definable equivalence relations on Polish spaces.

Namely, an equivalence relation $E$ on $X$ is Borel reducible to a relation $F$ on $Y$
($E\leq_{\text{Bor}}F$)
if there is a Borel measurable map $f:X\to Y$ such that for $x_1,x_2\in X$, $x_1Ex_2$ iff
$f(x_1)Ff(x_2)$.

Now, if the isomorphism relation is as simple as the identity on the reals,
then the objects in the class have a "simple" classification.

If the relation is not as simple as that, but for example as simple
as "two sequences of 0's and 1's agree on a final segment", then the classification is more difficult, but still not very difficult (since there are more complicated isomorphism relations).

Names connected to this approach to classification are Kechris, Hjorth, and Su Gao.