One exception of the "keep one ball" practice that comes to mind is Serena and Venus Williams. Neither of them ever keep a second ball on them. If their first serve results in a fault, then they will request a second ball from a ball person. You can see this if you watch their match footage.
For the players that request 3, 4 or 5 balls (there is no limit to how many they can request and there are typically 6 balls in play at any given point during the match) - what are they doing to decide which ones to give back to the ball person and which to keep for serving?
It has to do with the look of the ball mainly. The fuzz on tennis balls is very dense and matted down when the balls are new and unused. As they are hit, the fuzz becomes less and less matted down and "fluffs up" - and therefore less aerodynamic, causing the ball to travel through the air slower. As a tennis player - when you're serving, you want the ball to travel through the air as fast as possible, so you're looking for the balls that are the least "fluffed up". This is why you will sometimes hear the commentators of tennis matches refer to new balls as "helping" the first person to serve once new balls are introduced into the match. This is also part of the reason behind the "new ball" gesture players give each other when new balls are introduced. This gesture is made to the player returning serve by the player serving as sort of a "heads up, the balls might fly faster now" warning.
During pro matches, 6 balls (2 cans) of new balls are given to the players before warm-up and then new balls are introduced after the first 7 games, then every 9 games thereafter.
If you really pay attention to player habits and superstitions, you can notice some players won't even look at the ball given to them by the ball person and immediately hit it to the opposing ball person in the other corner because they want each ball person to have an even number of balls in their possession so they can turn to either ball person and know they can get 2 balls from them and not waste time/focus turning to a ball person just to see they aren't holding any balls.