In many ways, music is only one of many fields through which students may gain various important lifelong skills. Some people claim music to be more important than other fields such as athletics, visual arts, etc. However, in many respects, the best music teachers, the best athletic coaches, and the best science teachers are all essentially teaching how to be a dedicated, critical, and creative person. Music happens to be my field of expertise, so I teach through music.
Music's intrinsic values and functions should be paramount. Still, there is nothing wrong with discussing some of the more broad benefits as well. It is not reasonable to study music just because it helps with math, but some subjects, such as confidence, have no direct course of study. Confidence is not an activity or field of study, it is something gained through many varying activities, music among them. And while the best way to excel at math is to study it directly, it is alright to note that music does involve math as well.
Below is a list of extramusical items that can be learned through well-taught music study. This list includes both broad skills as well as tangential, interdisciplinary subjects. Any good music teacher should strive for including all of these. (And with such a broad list, there's no excuse for artificially connecting truly unrelated things to music just to be "interdisciplinary") This list helps define some of the what and why of music study, which should always be a prerequisite to studying the how of teaching or learning anything.
Music study may involve or encourage, in no precise order, explained only when I felt necessary (and not an exhaustive list):
Self-awareness & reflection
Imagination - specifically developing control of aural imagery ('audiation')
Creativity - through interpretation, composition, improvisation, and experimentation
Respect - for traditions, for teachers, for audience, for fellow musicians...
Teamwork - through ensemble playing
Independence - practice is often done alone, especially at more advanced stages
Vocabulary / language / poetry - through song and songwriting, and through analogous pitch and rhythm aspects of relating language to music
Control over one's own mindset - such as the ability to choose when to have one's listening or playing be more emotionally versus more intellectually focused
Control of the experience of time - different music can encourage smaller or larger time-scale focus. Also, learning to focus on longer and shorter sections of the same music is fundamental to advanced playing. Patience is actually a skill contained within this overall time-flow control.
Habit control - to succeed in music, one must understand of how habits work, how to recognize them, how to unlearn bad habits and develop desired ones.
Physical control, coordination, dexterity - some variance depending on instrument, but the physical skills of playing or singing are substantial. Good music teaching informs students what physical positions are healthy, how muscles work, and how to be conscious of these issues.
Pattern recognition - a major aspect to understanding and learning music. Awareness of the way human psyche processes patterns, including illusions is important.
Mathematics - unfortunately, typical western interval names in music (e.g. "major second") actually are mathematically problematic because they create a fence-post error (two major seconds make a major third, 2+2=3?!? — this is because the naming system lacks a zero and double-counts the note shared by both "second" intervals). Additionally, rhythm terminology is mathematical but often taught purely by rote. It is enabling to learn the real math behind how rhythm, harmony, and melody operate. Explicit math is not a necessary factor in music, but music is overall very mathematical.
Physics - the acoustics of instruments, rooms, and anatomy of the ear are all relevant and should be taught from the very beginning. I haven't yet had a student of any age struggle to understand that strings vibrate - and that is what makes sound. And furthermore that they vibrate faster and slower based on (A) mass (practically described as a string's thickness/density and length) and (B) tension. Many music teachers unfortunately deprive their students of these simple insights by simply asking them to accept by rote that a sound is "low" or "high." Learning these insights into the physics of sound is not only interesting, but also provides the ability to think critically about the actual effect of technical choices in one's playing.
Psychology - music is nothing if not psychological. Because it is so multifaceted, music offers significant insight into psychology in general and everything else listed here exists within a psychological framework.
Culture and history - Music is among the most ancient elements of human society. Musical patterns that happen to have become common historically are very relevant to being a musician in the real world, but they need context and qualification. The best teachers should not say "music works like this" when referring to a cultural tradition. All music should be taught with cultural and historical context.