Consider the following pairs of statements:
"I see what I interpret as a chair" vs. "I see a chair."
"This chair can be interpreted as a set of atoms" vs. "This chair is a set of atoms."
"This set of atoms can be interpreted as a chair" vs. "This set of atoms is a chair."
"The universe can be interpreted as a mathematical structure" vs "The universe is a mathematical structure."
"The mind can be interpreted as a pattern of neural activity" vs "The mind is a pattern of neural activity."
"The group of rotations of a cube is isomorphic to S_4" vs "The group of rotations of a cube is S_4."
Let us consider in particular the last of these sentences, about the group of rotations of a cube. It's quite common in mathematics to speak this way. When we have shown two structures are isomorphic, this means that theorems (of a certain kind) that we deduce about one of the structures can be directly translated into theorems about the other structure, and vice versa. A mathematician thus will casually refer to the two structures as "the same," even though technically there is a difference in how the structures are defined. They are the same for his purposes.
An isomorphism in mathematics is a kind of rigorous interpretation; we look at one structure and interpret it as a second one. And from then on we can treat them interchangeably, and comfortably refer to them as the same structure, as long as we restrict ourselves to propositions of a certain kind that can be "translated across" the interpretation in either direction.
The other statement pairs listed above are similar to the statement pair about S_4. In each case, there is an object, and an interpretation of the object, so that statements of a certain kind made about the object translate into statements of a certain kind about the interpretation, and statements of that kind made about the interpretation translate back into statements about the object.
So, is it reasonable to say that the statements in each pair are equivalent? In other words, to use the word "is" in a manner that does not denote exact equality (whatever that is), but that denotes equivalence of structure with respect to a certain class of statements?