Hello, as you know restoring the multiplayer games in the last version of Age Empires II (2013) got broken. Simple, you can't load the saved multiplayer game. The solution is to downgrade AOE 2 (2013) to the patch where that feature was not broken.

How-to:
1) Go to Steam Library, find "Age of Empires II (2013) and right click on it then choose "Properties..."


2) On the opened window click the tab "BETAS" and select the beta "escapeaoe-5.2 - EscapeAoE 2 Tournament".



3) Close the "Properties" window and wait when Steam downloads old version. Then enjoy.

P.S. Be careful, it does not allow to restore the saves from the other versions, so you have to play in this beta version from begin and thus you can restore the games.