Many companies that made wedding sets "customized" their products with safety or comfort wear mechanisms. Is their a manufacturer name or hallmark inside the ring ? If so, you might try to contact them for information. It's possible that the metal is a tab that when the ring is worn the finger pushes it in and it automatically locks to the second ring. Another idea is to take it to your jeweler and maybe with magnification they can determine how the mechanism works.
I am quite familiar with this old style design with the "slide-in slide-out" little arrow looking thing designed to keep the rings together.
The device is an arrow head looking end on a piece of metal which slides from one ring side. The other ring has a side which receives the little arrow. Sometimes, you have to slightly twist the other ring to get the metal point to go in and stay in place. This is not a permanent connection, only for when wearing the rings.
Over time, the slide connector and the connector area of the second ring will wear. After many years, the connection simply refuses to work properly. A good cleaning by a jeweler with both ultrasonic and steam may loosen the connector so it will work and clean out the receiving end of the other ring. This might help. If worn, the best bet is not to even try to use it and wear the rings "as are"
Repair would be difficult if the connector is badly worn on either side. Rather than that, one option is to have the rings spot soldered together at the bottoms of the ring bands, for a permanent join. This means both rings are always worn together and there is certainly no problem of one ring twisting from the other when worn. You may or may not want to do this.
Of course, a jeweler can simply remove or tack the little slide part in place so it does not slide in and out, keeping the rings separate with no connection. If the rings stay on your finger in a pleasing way, you do not need the connector to function in the first place. You can just wear the rings "as are"and not be concerned with the connector at all. Just wear as two rings together on the finger and don't worry if connected or not. This means no work to the rings at all and is the easiest way to resolve the use of the connector.
These devices were popular those years ago but have not been seen in wedding sets for almost as many years! Apparently the advantage of having the connector never really did not prove to be a reliable and viable function...or was one not really needed when the rings were correctly sized to the finger. The connector is almost a "relic" of ring design from many years ago.
Try to get the rings cleaned well, see if the connector will fit into the other ring (check both sides to see if there is one opening different from the other), if so, wear it a while and see what you think. Overall, I would be more concerned with the condition of the stone settings at this time. The rings should be checked to be certain years of wear have not left any stones with thin and insecure settings.
That is the first priority. Secondly, play with the connector and if it works it should work easily, sliding into the opening in the other ring and holding things pretty much together when worn. If it does not work ok, simply forget about it and keep it as a curiosity of years gone by, a touch not seen in modern ring design for whatever reason.