I like to think that the prince is a master strategist. They had a ball and invited all the girls in the kingdom. This lets the prince see what all the girls look like when they're dressed up and how well they conduct themselves at social functions, but that's not actually all that important. What he needed, after determining whether a girl could pass muster as the public face of the monarchy, was an excuse to make a surprise visit to all of these girls to see what they look like when they're not doing their best to dress up for him, because he's a smart enough dude to know that makeup and fancy clothes are how women lie to you that they're attractive when they really aren't, and a monarch doesn't stay in power for long without double checking everybody's lies. The shoe is just an excuse to make these surprise visits. If that hadn't happened, he'd have found something else.
At this point, he doesn't actually care that much about Cinderella. She's just some girl he met at a party, one of a few who managed to not make fools of themselves. The prince is keeping his options open and checking out all the girls in the kingdom when they aren't ready for him, to see what they really look like. That's just gravy for the Prince, but what he's really doing is testing for initiative and cleverness. The glass slipper is tiny and they aren't exactly being careful with it, so breaking it isn't hard. A sufficiently crafty girl could see that they wouldn't be able to fit into the slipper and "accidentally" kick it off of that poofy pillow they were carrying it on to guarantee its destruction, and then argue her case that she was the one. Merely making the attempt to do this would show initiative (and a willingness to lie, which is an essential trait for a ruler), but nobody else even thought to try it.
So, Cinderella's evil step-people break the slipper, and Cinderella presents a replacement which fits. Note that there is no way for them to check that the slippers are actually the same size, and nobody cares about that because it's irrelevant whether a girl would actually fit into the slipper that was left behind. What matters is whether a girl would be smart enough to break the slipper that was left behind and then make the case that they were the mystery girl from the ball. The prince was probably under the assumption that Cinderella had her family break the slipper on purpose, which displays initiative, cleverness, leadership, and the willingness and ability to lie convincingly. At this point, the prince is actually smitten with her and can hardly wait to take her back to the castle and start ensuring the royal succession, but he plays it cool to see how she's going to wrap things up.
Cinderella pulls the other slipper out of her dress. The Prince figures this is all just going according to plan, because who carries a single glass slipper around the house? Nobody, unless they knew they were going to need to present it as evidence, and Cinderella showed up a little late for the shoe-fitting event; the prince assumed after the fact that she was going to get the other slipper, to make the big reveal more dramatic. That's good enough for him. He marries her and is disappointed when he discovers that she's not actually a master of deception, but her ability to speak to mice and enlist the aid of birds makes her an unexpectedly useful asset in running the kingdom's spy network, so they have their happily ever after anyway.