The rings are the symbol of marriage. When Portia gives Bassanio her ring, she charges him not to give it away or lose it. For her, it symbolisms her committment and love to Bassanio.
Leah was Shylock's deceased wife. For Jessica to give it away for a monkey is a stab in Shylock's heart since he received it from Leah. If she had wanted to hurt her father, she did by this action. For Shylock it was a precious memeory that she has given away.
Are the rings related? In the fact they they are the symbols of love and marriage, the simple answer is yes they are.
regarding kutiepie's Q from 8/16: Portia's lengthy speech where she gives Bassanio the ring(3.2) tells us something about what Leah's ring signifies, I think, as both were given before marriage. In Romeo and Juliet, Juliet has the nurse deliver a ring to Romeo(ROM3.2). When Romeo bids his man farewell, he says that his chief intention is to retrieve "A precious ring"(5.3).
I don't know of any significance it has to Portia nor Nerissa's rings. However it is to my understanding that the turquoise is an eastern talisman which, in the west, was held to cause sterility. Therefore it stands to reason that Jessica may have ddliberately parted with the ring.
Source: M. M. Mahood (editor), The New Cambridge Shakespeare, The Merchant of Venice, 1989