The Style Manager lets you create Maplex Label Styles, which define how a label will look and placement properties etc., but will not save the label definition (i.e. what field is labelled).
To create a new Maplex style, all you need to do is right-click in the empty space alongside where it says "Maplex Labels" and select New > Label...
You can copy label settings from one layer to another by using the Label Manager from the Labeling toolbar:
On the layer you want to copy label settings from, right-click and select Copy Parameters, then on the layer you want to copy them to, right-click and select Paste Parameters. This will copy all the settings such as style, field name/definition, placement.