QTreeWidgets are independent of each other with separate data and selections. A shared selection is only really sensible if the underlying data model is also shared.

If you want the same data presented in three view then you should use QTreeView and a separate model. You can build a usable tree model with QStandardItemModel or write your own custom model.
If you have three views sharing the same model and want the same current and selected items in each then point them all at the same QItemSelectionModel (i.e. share one).