Maybe you’re confused because the etymology is not clear to you. I see the term “wedding ring” and I think it denotes that person A and person B are being wed to each other. Joined to each other. The ring symbolizes that joining. In other words, I don’t think the ring is named after the event. I think both the ring and the event are named after the verb. In which case it’s a very normal name.
Another interesting point is that in some cultures it’s quite common for people to get married months or years before they have a wedding. They can go to City Hall and get married, and then later when they have time and money they can schedule their wedding celebration party. If that’s the case, then the ring that you use for the first time at your wedding might be reasonably called the “wedding ring”.