I had the same question. I had no idea that each team plays with their own set of footballs.
Now that I know, I am left with the thought, “What an incredibly stupid idea. Why not use one set of footballs for both teams to ensure parity?”
RE - one set of footballs. What I’ve heard/read is that each team has a supply of footballs with their own team logo on it - makes them much more valuable that way. Also they do not use brand new footballs on game day. They use footballs that they’ve used during practice which they are permitted to scuff up within reason. The balls don’t have to be new, they just have to pass the pre-game inspection.
...they used to. Manning and others lobbied for their own balls so they could get the grip they want. Kicking balls are another matter, as they are shared, and nearly new.
The main reason they don’t is apparently because this fuss is just silliness in a gotcha that no one cared about because it doesn’t actually matter after about 10 pSI.