Being the type of person who likes to relate mathematics to everything, I think of an isomorphism as being isomorphic, so to speak, to the literary or linguistic concept of an analogy, metaphor or simile.
One of the primary purposes of an isomorphism in math is being able to study the structure of something in a different context, and this is also the purpose of things like analogies. We can compare or "map" the attributes of an intangible concept, say, to a physical object to try to get a better understanding of its nature by studying something that is similar in ways but easier to grasp.
Of course, the isomorphism will eventually break down. This will always happen when one tries to find an analogy between the perfect, crystalline mathematical world, where everything is ordered and idealized, and the real world, where things are random and uncertain. Similes and their like are not symmetric; if A is a good simile for B, that won't mean that B is a good simile for A. They are also not transitive. If A is a good analogy for B, and B seems to be analogous to C, it doesn't mean that A will be a good analogy for C.
The basic concept is the same, though, no matter how imperfect the translation. It's often helpful to try to find isomorphisms between mathematical concepts or other abstract ideas and more familiar and tangible things, because it's easier to understand something when it looks like something you already know.
Thanks to Gritchka for advice on the mathematical properties of similes etc.