It's called the "one ball condition" (in the rulebook, it's in Appendix 1, Part C), USGA rules.
The "one ball condition" requires the player to use the exact same brand and type of ball throughout the round. For example, if you tee off the first hole with a Titleist Pro V1x, then that's what you must play throughout the round. You may not switch to any other brand of ball, nor even to any other type of Titleist ball. You started with the Pro V1x, so the Pro V1x is what you must use on every stroke.
In real golf if you ran out of the same type of ball you would be disqualified. For this game it switches you to the starter balls out of courtesy to finish your round. The game also gives you the opportunity to purchase the same type during the round. This game conforms to the USGA rules so they are the Village Idiots that wrote the rule it as you put it.
During your previous round it shows if the ball is getting low and turns red on the life scale. So if you start a new round with that ball you better have more of the same ball or you will more than likely revert to the Starters during your round. Try playing Practice rounds or use the single hole play via the Course under WGT Community to try out new balls and clubs if you don't want to have to end a round.