You can't continue to play a basketball game w/ a spare basketball staring at you through a glass backboard. It would be a bit distracting to say the least.
Some college coach in the 30s or 40s had the equipment manager sew footballs to all the teams uniforms, so it was, "part of the uniform." But it was really to confuse the other team's defense. I think the opposing coach had the football died. I am pretty sure the league disallowed it after that.