Do you mean why are the objects used in games ball/spherical shaped, or why is an object used at all? Either way I'm not sure how much of a history question this really is, but I'll give it a shot.

Regarding why spherical, it offers a lot of advantages for a sport. The relatively flat even shape allows for reliable and predictable movement when tossed or kicked. The sphere shape allows it to roll and bounce along without having to always be carried. Ball games that use oblong balls usually rely on throwing the ball more than kicking or launching it by some other means.

Regarding an object of possession (typically a ball) there are plenty of games that don't use them. Badminton, darts, racing, fencing, skiing, swimming, martial arts and so on. The common theme is that these are usually individual sports. What you likely find is that most team sports use balls because it serves as a central point of focus in games with many people. Trying to organize a sport with large teams playing against each other usually results in lots of smaller matches within the central game. Think of a huge race, you're only going to watch the front few people. Or a big wrestling mach is just chaotic and as soon as one team has a man advantage they will dominate the rest of the game. A single ball allows all players to work on a single goal together. This is also likely why ball based games are so popular - they are easy to follow while spectating.