What!? Did the developer of Stardew Valley use Tiled to create the maps, or is there some converter / importer available somewhere?
Ok, a quick search yielded these instructions, which seem to indicate he used tIDE, but tIDE can export to TMX, so I guess this is what you did? It may be interesting to add
.tbin support to Tiled so the conversion step isn't necessary...
In any case, I have currently no idea what may be going wrong with the copy & paste. Are you having these issues only with these Stardew Valley maps? If so I would need to try opening those in Tiled as well to find out what's going wrong.