"explain by analogy, exhibit resemblances between," 1650s, from analogy + -ize, or else from French analogiser (17c.). Greek analogizesthai "to reckon, sum up, calculate, consider" suits the form but not the sense. Related: Analogized; analogizing.
early 15c., "correspondence, proportion," from Old French analogie or directly from Latin analogia, from Greek analogia "proportion," from ana "upon, according to" (see ana-) + logos "ratio," also "word, speech, reckoning" (from PIE root *leg- (1) "to collect, gather," with derivatives meaning "to speak, to 'pick out words'").
A Greek mathematical term given a wider sense by Plato. The meaning "partial agreement, likeness or proportion between things" is from 1540s. In logic, "an argument from the similarity of things in some ways inferring their similarity in others," c. 1600.
word-forming element used to make verbs, Middle English -isen, from Old French -iser/-izer, from Late Latin -izare, from Greek -izein, a verb-forming element denoting the doing of the noun or adjective to which it is attached.
The variation of -ize and -ise began in Old French and Middle English, perhaps aided by a few words (such as surprise, see below) where the ending is French or Latin, not Greek. With the classical revival, English partially reverted to the correct Greek -z- spelling from late 16c. But the 1694 edition of the authoritative French Academy dictionary standardized the spellings as -s-, which influenced English.
In Britain, despite the opposition to it (at least formerly) of OED, Encyclopaedia Britannica, the Times of London, and Fowler, -ise remains dominant. Fowler thinks this is to avoid the difficulty of remembering the short list of common words not from Greek which must be spelled with an -s- (such as advertise, devise, surprise). American English has always favored -ize. The spelling variation involves about 200 English verbs.
<a href="https://www.etymonline.com/word/analogize">Etymology of analogize by etymonline</a>
Harper, D. (n.d.). Etymology of analogize. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/analogize