Doesn't have to be a religious item. The big medieval "take home relic" was ordinary handkerchiefs or strips of cloth. Touching a relic is a higher class of blessing than having something blessed by a priest. (Though obviously, you would still want a relic-touched rosary blessed by a priest, to be eligible for certain blessings from the church.)
So yes, the rings would now be third-class relics as well as blessed wedding rings.
Beyond that... blessed wedding rings are a sacramental by virtue of being blessed (especially if as part of the Sacrament of Matrimony). They are definitely religious items now!
That said, make sure that you remember that they aren't ordinary rings anymore. You can't just throw them away; they must be burned or buried. You can't sell them to anyone; you can only destroy them or give them away. You can certainly have them resized, cleaned, etc. without losing either their sacramental or relic qualities, however. :)