soft question - Analogies between analogies

It is unfortunate Jan Weidner asked his question -- "Can someone give an example of an analogy between analogies?" -- in such a way that he is unlikely to receive very many insightful responses of the kind offered by Joel David Hamkins and philip314. I don't know how insightful my own response will be, but I can provide a context for the topic of analogy, as I was the contributor who added the quotations by Stefan Banach and Stanislaw M. Ulam to the list of famous mathematical quotes.

It is true that Banach is the original source of the quotation, but I first encountered the topic in Ulam's memoir Adventures of a Mathematician (1976; 1991). From there I learned that the University of California Press published a collection of papers by Ulam titled Analogies between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators (1991). The collection includes the paper "On the Notion of Analogy and Complexity in Some Constructive Mathematical Schemata" (1981). Rather than attempting to summarize Ulam's results, I'll simply provide an excerpt from the paper's introduction so that interested readers can investigate the topic on their own:

Throughout the development of
mathematics and with the growth of new
concepts and more complicated notions,
a cohesive tendency and organic
structure have been guided by a
feeling of analogy between the old and
new ideas.

Historically, problems posed by the
development of a new mathematical
discipline, which originally was only
metamathematical, coalesced into new
parts of mathematics itself. One could
cite, as obvious examples, the study
of transformations of a space as
points of a new space of such
transformations, and the study of
algorithms for solving equations as
entities per se (group theory, for
instance).

The increasing proliferation of
notions in pure mathematics may
suggest that the idea of analogy
itself is amenable to mathematical
discussion. One finds that old and
elementary formulations of this idea
are, in special cases, present in the
definitions of the similarity of
geometrical figures, more generally in
the equivalence of figures-sets,
through the elements of a group of
transformations, or, more generally
yet, through the identity of proximity
of such sets in spaces which encompass
them.

Two abstract sets of elements may be
felt to be "analogous" if the
difference between their cardinalities
(in the finite case) is small compared
to the cardinalities themselves. Two
classes of such sets may be deemed to
be analogous if the numbers of sets in
the two classes differ by "little" and
if the cardinalities of the
corresponding sets also do not differ
by much. Obviously, one needs to
attempt to formulate a quantitative
criterion, and it is clear a priori
that the notion of analogy will not
be, in general, transitive.

In this report we merely want to
discuss some of the salient features
of analogy and exhibit them on a class
of examples where we shall attempt to
define it, at first, as proximity in
the sense of a metric distance in
suitably defined spaces. (Ulam 1991; 514)

Hopefully, Ulam's paper will provide some tools to make the topic of analogy less "far out."