Literally speaking, a model, like, airplane is isomorphic to some degree or another to the real airplane of which it is a model. Is that how a scientific theory works? Is there some isomorphism between the parts of the theory and things in the world?

Yes, just like that.

In science, a model is a set of variables that stand for physical quantities, together with a set of relationships between those variables, which are asserted to correspond with the relationships among the physical quantities. The relationships are typically expressed mathematically.

For example, s = (at^2)/2, where s is the distance travelled in time t by an object under constant acceleration a starting from rest. This is a model of what happens when you drop something.

More generally, there is a Wikipedia page, which is sound but I think over-complicates the idea (and the section on "Business process modelling" doesn't belong there at all), and even more so the disambiguation page for "Model", but the same fundamental idea runs through the whole.