I'm not sure if the Swan & Dolphin participate in this program, since they are not owned by Disney.
With the exception of the All Stars & Port Orleans, mugs are not interchangeable at the resorts. What I mean is that you cannot use a Contemporary mug at the Wilderness Lodge & so on.
The refillable mug program has been a hot topic, with people having opinions from both sides. I have a collection of mugs going from the different resorts we have stayed at and yes, if I am staying back at one of these resorts, I bring my mug from that resort with me from home. I have asked several times and have always been told that this is not a problem, since it is from that resort. Now other people are going to disagree with this and they can, but I have always asked before bringing it and again, have always been told no problem.