The stroller rental CMs don't want to see or swipe your park ticket and they don't need to. You just tell them how many days' rental you want. You'll pay for it all up front minus a 10% discount. They'll give you a strip of paper tickets stapled to your receipt.
Don't lose those tickets, because they are NOT replaceable. If you lose 'em, you'll have to pay to replace 'em. If it wasn't that way, it would be WAAAY to easy for someone to say they lost their tickets, get free replacements, then sell the originals or the replacements.
The stroller rental place at each theme park has two lines, what I'll call the "pay" line and the "stroller" line. When you pay for your multiple day tickets, get in the pay line. Once you receive the tickets, take one to the stroller line (you don't need to detach it from the strip, and I'd recommend you don't, to promote not losing it). At the front of the stroller line, the CM will date-stamp one of your tickets, give you a large card to write your name on, and give you a stroller. You place the card into a plastic sleeve on the back of the stroller, to aid in identification (pretty impossible to tell one rental stroller from the next!).
When you return your stroller, you can hand in your date-stamped rental ticket for a crisp Disney Dollar. BUT, if you're returning to the park that day or if you're hopping to another park later in the day, don't hand in your date-stamped ticket -- you'll need it to get your next stroller later that day. Actually, I never turned in my rental tickets until my last day and when I handed them all in I got a whopping six Disney Dollar windfall.
We never took the writing our last name on the card too seriously. Each time, we had the kids pick a character name and I wrote that on the card. One time we were Buzz, one time we were Goofy (well, we're ALWAYS Goofy

), etc. If you don't turn in your date-stamped ticket when you return your stroller, it's kinda funny when the CMs tell you to take your name card with you to use at the next park. Yeh, like I'm going to bother carrying it from park to park, like I can't make another one!
When you get another stroller ANY TIME AFTER YOUR INITIAL RENTAL, totally skip the pay line and go right to the stroller line with another unstamped ticket. Skipping the pay line is the other advantage to multiple-day rentals, beyond the 10% discount.