Judgments that two things are similar are very important in human reasoning. There are many kinds of similarities, and three important ones are the following:

• Similarity of A and B because A and B have one or more properties in common
• Similarity of A and B because A and B have (approximately) the same number of elements
• Similarity of A and B because one is supposed to be a sample and the other its population.

The first kind of similarity - having one or more properties in common - is the most general, and one to which other kinds of similarity reduce, but it tends to be not helpful, because it is evident that all pairs of things, however many different properties they have, also will share some properties.

The second kind is treated by mathematics. See also: Isomorphism, Morphism, Frege's Theorem. In general, the most interesting analyses that have to do with the notion of similarity (and related ones) are mathematical.

The third kind is treated by statistics, but also done automatically to some extent by one's brain, that's quite good at picking up patterns and (ir)regularities in a statistical sort of way.