Snapchat gave me the real name of a user I dont know. How? A long time ago I added a person to Snapchat. His username (display name?) was O*, and I added him with that display name. Today I received a message from him, but the name that showed up in my notifications was S* R* (his real name). When I opened the message, his regular display name O* was displayed.

What can explain that his real name showed up in my notifications?
It may be a bug in Snapchat. You can send snapchat a contact form explaining how it happened.

But you cannot be sure it is the real user name, unless you asked.

Since it is Snapchat, I suppose you shouldn't. Just ignore the name you saw, and go on with your usual business.