Since Lollipop it seems like the background of the menu you reach when pressing the "menu" bottom at the middle of the bottom of the phone is not the desktop background anymore but rather is just white. Can I somehow set it to be again the desktop background? This is Android 5.0.2 on a Moto G (2nd gen)
From what I'm reading, this must be something they took off for Android 5.0 Lollipop, as it sounds as if you need a custom ROM, or a special launcher to make this happen again. It no longer supports having the background image on the menu.
In Android 5.0 Lollipop, Google introduced a white background to the app drawer screens. This follows their material design philosophy, but may be too bright for some of us.
There is no setting to switch app drawer themes or get KitKat's dark semi-transparent background back, but you can change this by installing a different launcher app.