The term "audio engineering" can refer to several different fields. Some engineers work in the recording industry. Recording music involves many people with engineering skills related to sound: recording engineers, mixing and mastering, and others. These professionals tend to have backgrounds in music technology (a field that you could get a masters degree in), or in electrical engineering or other related fields.
Then there are engineers who work for companies that make loudspeakers and headphones and other audio technologies. Companies like Bose, Harman, Sonos, and many others employee large numbers of engineers that develop designs for new audio technologies.
There are audio engineers who do "live sound" - people who design and set up and operate the sound systems for events like theater and music performances.
And then there are sound system designers: professionals who design the audiovisual systems that get installed in performance spaces, classrooms, corporate boardrooms, and elsewhere.
And then there are people like me: I work in architectural acoustics. I help architects to design buildings that sound good - sometimes that means controlling noise from mechanical equipment; sometimes that means isolating sound from the outside; sometimes that means shaping a room and using sound-absorbing and sound-reflecting finishes to create the right kind of acoustic environment for a space - whether that space is for music, for speech, for learning, or for some other function.
These fields all relate to sound, and they are all wonderful, necessary, and rewarding. The people involved tend to have diverse backgrounds: some come with a music background, others electrical engineering, others mechanical engineering, some physics, some architecture, some other design-related fields. The key is a willingness to listen actively and critically to what you hear in the world around you.