Unless it's recently changed, you don't get a card and receipt to show each day.
In order to get a stroller, you show a small piece of paper, a rental pass. That rental pass must have TODAY'S date stamped on it (OR it must have NO DATE stamped on it, and then the CM will stamp today's date on it).
But I'm getting ahead of myself ...
If you will be renting park strollers for more than one day, then on your first day you will get into the payment line at the stroller rental place at any of the 4 theme parks. You tell the CM how many days' rental you want. You pay in advance for that number of days, and yes, there is a small multi-day discount. The CM will give you a strip of perforated rental passes, stapled to your cash register receipt. The CM will stamp ONE of the passes with today's date.
Then you go to the line (there may not even be a line) where you actually receive your stroller. You show the CM the rental pass that is stamped with today's date. You don't need to remove it from the strip of passes, you just show it to them. Then you get your stroller.
Nothing happens when you return your stroller when you leave (no rebate, they don't take your rental pass or even look at your paperwork, etc.). If you go to another park later that same day or you return to that same park later that same day, you skip the "pay" line and go right to the line to get your stroller; you show the CM your same-day-stamped rental pass, and you get your stroller, easy as pie.
On subsequent days, you totally skip the pay line and go right to the stroller receiving area. The first time you do this each day, a CM at the receiving area will stamp a blank rental pass with today's date on it, etc.
The ONLY real reason you'd need your cash register receipt is to get a refund. Unused stroller rental passes are fully refundable, but you need your original cash register receipt along with any unused passes in order to get the refund.