First, you've got to get over how remarkably silly coaches would look in basketball, hockey, or football unis.

(Although George Carlin does a good bit on this...including wondering why so many Other Sport coaches wear baseball caps when there's no danger that a baseball coach would choose, say, a football helmet --- with the possible exception of Zimmer.)
In the early days of baseball, the manager/coaches were frequently players. The only manager, I believe, who didn't wear a uniform was Connie Mack, and by that time, the tradition was so firmly entrenched that he wasn't allowed on the field in his suit.
So now it's tradition and practice, though I'd also add 'because they can'. It is a set of clothes that doesn't preclude anyone because of age or shape.