Boris,
The latest version (1.8.1) of Custom Maps no longer can store the maps in the public internal storage due to Play Store rules.
But you can export all the maps using the "Export all maps" menu item in the maps list screen. At least Google Drive can receive the maps that way.
To import the maps onto your new phone, you can use many file managers (at least Google's "Files" app works, but I know Samsung's "My Files" does not recognize .kmz map files). When you select to open the .kmz map files with Google's "Files" app, it should offer to you to open them using Custom Maps.