I think you did the right thing. Mistakes happen, and very little good would come out of demanding that this person gets fired as a result of one (though the employee probably did deserve a bit of embarrassment). The manager probably should have apologized, but it's not the end of the world.
I will admit that I get a bit paranoid with those bag-check things at the stores here. At the stores in my neighborhood, they know me well enough to where they almost never even check my card, it would be super easy for me to grab someone else's bag. I've never done it, I don't want to do it, but I could...that's why I usually don't go directly before or after work, because I don't want my bag with an expensive Macbook to get stolen.
Where I live, supermarkets have a reception where you can leave your belongings while you're shopping. The employee in charge gives you a numbered card that matches the drawer in which they stored your things (all of this is handled by the employee) and off you go. No keys or anything, only a drawer with a card.
A few months ago I left my backpack at a one of these. 20 minutes later, as I was about to leave, I went to return the card in order to retrieve my stuff and to my surprise they gave me someone else's backpack.
I kindly asked what happened and after the manager had gone to check the CCTV footage they told me they gave my backpack to another customer. The problem? The numbered card didn't match the drawer and no one had realised. And there I was with the stranger's backpack and no way to contact him.
I agreed to wait until this man came back to return my stuff so we could switch backpacks (he had to; his work ID was on his backpack). I went home, told my family what happened, and they asked me what I told the supermarket staff. "Nothing, I'm just gonna wait" I said, and they rambled on about how I should have yelled at them and made a scene and maybe even made them fire the employee that grabbed my backpack.
I didn't do any of that because I precisely didn't want them to fire the poor minimum wage worker. Besides, I understood all of this happened because the system of numbered card/open drawer is completely broken. It wasn't relly the employee's fault; although it was his mistake not checking the numbers not matching. Yelling wouldn't have fixed anything.
I got my backpack back the next morning. No one was fired. The supermarket manager didn't even apologize for the inconvenience though, but ok.
Be kind.