Consider the following example:
Let $A$ be a set with more than one element. Let $\preceq$ be defined on $\mathcal P(A)$ as follows: $x\preceq y\iff |x|=|y|$. Now all the singletons are isomorphic, all the pairs are isomorphic etc.
We can use this for some definitions, too. We say that $A$ is Dedekind finite if there is no $x\in\mathcal P(A)$ such that $x$ is isomorphic to $A$. Otherwise we say that $A$ is Dedekind infinite.
Another example is this:
Let $A$ be an infinite set. Let $P$ be the collection of all partially ordered sets over $A$. We now say that $R_1\preceq R_2$ if $R_1$ embeds into $R_2$. For example, over a countable $A$ there are plenty of non-isomorphic orders which are isomorphic in this sense, e.g.
Let $R_1$ be isomorphic to the rational numbers and $R_2$ isomorphic to the rational numbers in the interval $[0,1]$. These two orders are not isomorphic as orders, but clearly bi-embeddable. So $R_1$ and $R_2$ are isomorphic in the sense that they are equivalent in the pre-order.
On the other hand, $S_1$ being isomorphic to the natural numbers and $S_2$ being isomorphic to the integers are not isomorphic in the pre-order sense, $S_1$ embeds into $S_2$ but not vice-versa.
This can be generalized quickly into plenty of examples, all of which have some use in mathematics. Such examples include groups with injective homomorphisms; compact Hausdorff spaces with continuous injections; etc.
Indeed the idea behind categories is to identify things up-to isomorphisms, rather than fully identifying them as sets.