Do different sports require different types of ankle braces?

Not really. Unless during a particular motion you feel discomfort with the particular brace you are wearing, as some are made out of a softer material and others have a harder "exoskeleton".
Not really. There is no sports-specific ankle brace, but different sports (and different positions within the same sport) lend themselves to different types of bracing depending on how much freedom in ankle motion is required. For instance, a football lineman might tolerate more ankle support than a gymnast.
No. There is ongoing controversy as to whether taping or bracing is more effective. The arguments are that taping is custom and better fitting but stretches out rapidly. Braces are generic but can be easily adjusted a d tightened as needed. If you have access to a professional trainer i recommend tape. If not, bracing is more reliable.