Can a party sue twice for the same money even though they have recovered those funds already from the first party sued?

I'm not a California lawyer, but I suspect that one of two things is going to happen with this.  One possibility is that the second suit will be dismissed, because it could have been brought with the first suit and therefore cannot be brought separately later on;  some states's courts are much stricter about this sort of thing than others.  The other possibility is that the insurance company that settled the first lawsuit will claim the right to whatever "A" wins in the second suit, up to the amount of the settlement, by way of "subrogation."

One way or another, if the court and the insurance company know what is going on, there will be no double recovery.