“Woman” is to “coquette” as “man” is to what?

tl;dr: So long as you are content with historical terms rather than merely contemporary ones, the precise word you are looking for is coquet.

Here’s why.

The OED says that a coquette is first:

1. A woman (more or less young), who uses arts to gain the admiration and affection of men, merely for the gratification of vanity or from a desire of conquest, and without any intention of responding to the feelings aroused; a woman who habitually trifles with the affections of men; a flirt. In early use the notion ranged widely from gallantry, wantonness, or immodesty, to pretty pertness.

Curiously, it also provides a later subsense:

c. male coquette: = coquet B 1

Where we learn that the sense B 1 of coquet is

B. sb. 1. A man who from vanity or selfish motives aims at making himself generally attractive to the other sex; a male flirt; a ‘lady-killer’.

Whence we learn of this curious etymology:

a. Fr. coquet, orig. sb., dim. of coq cock, in reference to the strutting gait and amorous characteristics of the cock; hence ‘a beau’, and in the fem. coquette ‘a belle’; also as adj. ‘forward, wanton, gallant’: cf. cock v.¹, also cockish, cocky a., and cocket a., which is prob. the same word in an earlier stage of meaning. In later use, the adjective in English became inseparably blended with the attributive use of the sb., to which also it became entirely assimilated in sense. The sb. was, as in Fr., formerly used of both sexes, and in both spelt coquet; but in the 18th c. the fem. became coquette after Fr., and the masculine became obsolete.

The pronunciations given for both are identical.

Perhaps one of the following may apply:

• rake
• ladies’ man
• player (urban and somewhat slangy)
• Don Juan
• Lothario
• Casanova
• wolf (less suave)
• womanizer

The term flirt works generically for both genders.

For a word that applies solely to men, philanderer is usually a good fit. You can usually find such terms by simply using a thesaurus.