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.