There are so many of them because they are phenomenological equations derived from exact theory by using any number of simplifying heuristic assumptions about statistical behavior, and there is a great variety of assumptions one can make in any given context, as well as a variety of different contexts. Even to "derive" the Boltzmann equation in kinetic theory through master equations, which is apparently a prototype for using them, one can come up with plenty, see e.g. On the Master-Equation Approach to Kinetic Theory by Kiessling and Lancellotti, who mention Kac's 1956 work on it as "pioneering".
The earliest mention in the MathSciNet is in Kac's entry to the 1954-55 Berkeley Symposium on Mathematical Statistics and Probability. The reviewer writes:"This paper is an attempt to justify and derive the Boltzmann equation by the use of the "master equation" [A. Nordsieck, W. E. Lamb, Jr. and G. E. Uhlenbeck, Physica 7 (1940), 344–360; MR0007519 (4,152b); A. J. F. Siegert, Phys. Rev. (2) 76 (1949), 1708–1714] which governs the time rate of change of the N-particle distribution function. The master equation is linear and further, incorporates Boltzmann's Stosszahlansatz [molecular chaos]". The use of scare quotes indicates that the term was not yet established then, and the cited Nordsieck-Lamb-Uhlenbeck article uses "master function" also with scare quotes:"If now a single electron of energy E0 is incident normally upon a plate of homogeneous matter, the nature of the emerging shower of electrons is specified by the "master function" giving the probability that, after crossing a thickness x of matter, any given number of electrons emerge in any given energy interval".
There are no mentions of "master" in the relevant sense before 1940, but the use of "master" in the review of Lehmer's 1939 article indicates that the word was used for something like "main" or "all-encompassing" rather than "masterful":"The automatic nature of modern card punching equipment is responsible for the remarkable fact only one error was detected in the master set for the new edition in the final comparison with the original edition". Brahana's 1940 article uses it in the same sense:"This group is called the master group for k, since every group of the class considered with the same k is a homomorphic picture of the master group". Translation of "master equation" to some languages literally means "main equation". This is plainly because it usually is the underlying mathematical device one works with, from which more specific properties and characteristics are derived, so it "rules" over them.