Copying All Settings Of One Composition in AE
This is not possible to do in one step, it is only possible to duplicate compositions:


Select the composition (Import Footage Panel)
Press Ctrl+D (or go to Composition > Duplicate Composition)
As (almost) everywhere in AE, hit Enter on the selected composition to rename the duplicate




When you need to copy the properties of a layer:


Select the Layer
Click on the tiny Triangle > left beside the layers name to open up the properties
Select Transform, Effects or Mask
Press Ctrl+C
Switch to the Second Composition
Select the Layer
Set the playhead to the animation start frame (first keyframe)   
Press Ctrl+V to overwrite all properties and insert all copied keyframes


To check the result, press U on the Layer, this will open up all animated properties of the layer. 

Note: Copying the same properties of different layers is also possible by selecting multiple layers, open up the properties simultaneously, and select them by holding down Ctrl or Shift. 



Related: After Effects: Shortcut for showing all transforming properties of a layer