Vibrato is a fundamental musical technique to enhance expression, particularly on longer tones.  Every melodic instrument has its way of utilizing it.  I think it’s helpful to think of vibrato both as a physical technique and as a cultural marker.  

Physically, you learn how to produce it on your instrument and it becomes a personal thing, part of your “voice” as you make your own musical statement.  In this sense, vibrato is a matter of personal taste and isn’t necessarily connected to a particular musical period or style.

Culturally, vibrato can be a stylistic marker.  Western classical music uses vibrato in carefully constrained ways, according to the performance practice of a particular historical period.  Just ask a Western classical player about vibrato in Baroque vs. Romantic music.  Or ask a jazz horn player about the difference between Dixieland jazz vibrato and Cool jazz.  As a clarinetist, one of my strongest jazz influences was Sidney Bechet, and in particular his broad vibrato.  Check him out, and then listen to the vibrato that a Greek clarinetist produces; even broader!  Check out this example from Stavros Pazarentsis: https://youtu.be/cOZ4j-hdlew.

One other thought about vibrato: Some music teachers are very cautious about letting students experiment with the technique because they feel it can make your playing sloppy and, even worse, it can become a habit.  This is true, but that’s true of most things in life; if you do them too much they become unconscious habits, rather than aesthetic choices.  Personally, I feel that it’s good to experiment, to study the use of vibrato in different styles and traditions.  Go crazy, test your parameters.  But then be able to make musical choices.  That’s always the mark of a more mature musician.

As a sax player I tend to use vibrato when I’m the solo melodic voice, and I tend to use it to shape the end of longer notes.  In certain jazz, blues and R&B moments, vibrato is a great effect to add as you fade out a note, and I often combine it with added breathiness, what some folks call the “foo-foo” sound.  Standard equipment for a jazz sax player.  Here’s a sample I did of Harlem Nocturne.  In my mind I was hearing the great alto sax player Earl Bostic, but I opted for a sweeter, more subdued tone, with just a hint of vibrato.