My understanding is that the use of the word "master"–as it tends to get used by US practitioners in US schools–is generally a European/US thing that started when these arts got imported after WWII, and one that varies heavily by style.
Japanese systems, at least, tend to just use the term "Sensei" (先生). Some systems may use another term for extremely well recognized individuals or founders of the system, who may be (and probably is) distinct from the instructor, but who gets that title is unique within the system and doesn't really translate cleanly to "master." Words like "Shihan" (師範) that frequently get called the title of "master" can have specific meaning and requirements of their own (independent from the dan grading system) and also is not cleanly translated to "master" with the same connotations. Just because someone is called "master" or "grandmaster" in a Japanese martial art in the US may just be a form of respect and does not necessarily imply that they have attained the title from the licensing organization or are actually an exemplary example of the art.
My experience so far has been that Korean arts tend to use Korean terms, some of which are based around their relative status (e.g., the owner of the school is kwan-jang-nim (관장님)), sometimes adapting these terms in the US to mean "Master" or "Grandmaster," depending. While if my research is right Muay Thai mostly uses "khruu" (ครู), which translates to "teacher" but in the US (or Europe) may still use the term "master," but it isn't immediately clear who gets the title.
Meanwhile, "sifu" (师傅), which gets used uniquely for a master of an art or craft (though not necessarily a martial art, and not for every profession), but the characters mean "teacher" and "tutor." The similar term that you might use for your own master as a form of respect (师父) takes the character "teacher" and combines it with the character for "father." The exact usage here is nuanced and, again, not something that cleanly translates into a US context that is applicable across different martial arts without an intimate familiarity with the cultural context.
Other systems have varying implementations on the theme, but the long-and-short of it is that it is going to be very art specific (especially in the United States), down to the details of the individual school, who gets called by what, and requires an understanding that the word "master" has a broad range of meanings that may or may not apply when you change cultural context.