I would not say that a manager should wear a uniform but a manager should look professional. For example, I like my managers to wear suits. The shirt with a collar and tie combination always exudes professionalism. However, this must be career appropriate. I would not have my management staff wearing a uniform such as the Best Buy's floor management or the McDonalds style shift manager's uniform.
In essence you must first look at your demographic. See to whom your managers will be representing themselves and ask if it is appropriate for your staff to wear a uniform, business casual, or business formal apparel. Then look to your employees. Ask what type of representation do you wish for them to see via your management staff. Then decide what you want your managers to wear.
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They wear uniforms because they sometimes have to enter the field during the game. In other sports they only do so in emergencies.
Basketball coaches wear coats and ties as a matter of decorum. A game is considered an important event and a coat and tie is appropriate for such an event.
Because they are a professional working as an employee in their chosen field of endeavor.