When I generate a city that I like, I add it to my collection thinking that I'm saving it for later use. However when I go to look at it in My Collection, all I see copies of a map generated by watabou that is not any of the maps that I added to my collection. What am I misunderstanding?

No, what you are saving is a link to the generator, not a generated map. Currently the only way to save a map is to copy and save a link in the non-itch version of the generator. For example this link http://fantasycities.watabou.ru/?size=15&seed=2129647494&hub=0&random=0&citadel=1&plaza=0&temple=0&walls=0&shantytown=0&river=0&coast=0 is a link to a specific map (Wibount Hamlet, 3 roads, a castle at the south edge).