Classification is the distribution of objects into groups. Systematics is the science of classifying biological diversity. The groups reflect biologists' best estimate of the relationships of the organisms.
Taxonomy is the naming of biological groups.
In the hierarchy of names above, each of the examples is a taxon. To biologists, similarity per se is not what allows recognition of groups, because not all similarity indicates relatedness. For example, whales and sharks both have fins, but many other features of their anatomy lead us to believe that they evolved fins independently. Biologists attempt to identify similarities due to descent from a common ancestor. Organisms can have primitive similarity because of distant ancestry, or derived similarity because of immediate ancestry.