What's another word for "perfect analogy" / "non-analogy"?

Is there a word to mean an analogy so perfect that it is no longer considered an analogy?

For example, what may be a suitable word in the sentence below? :

...The analogy eventually breaks down, as all analogies must. If it didn't it wouldn't be called an analogy, it would be called a _______ .

Solution 1:

An analogy compares two different things in order to highlight some point of similarity.

If the two things being compared are similar to the point of being identical, then you have a tautology -- describing the same thing using different words.

The first article referenced warns against pushing an analogy too far, explaining:

Just because two subjects have one or two points in common doesn't mean that they are the same in other respects as well.

In the nonexistent "perfect" analogy you're trying to describe, if the two subjects of the analogy are the same in all respects, then they are in fact the same subject -- and you once again have a tautology.

Solution 2:

You might refer to such a metaphor as a synonymy, which has an applicable sense (“(semantics) The quality of being synonymous; sameness of meaning”) as well as several senses that don't apply, or even are opposite of the intended meaning (eg: “(botany) The state of not being a correct name, of being [merely] a synonym”), or perhaps as an identity, in the sense “Any function which maps all elements of its domain to themselves”.

Solution 3:

The first word that comes to mind is "reality".