A layer and a feature class are two distinct concepts. A layer is a reference to a feature class. Layers in a map are configured independently of the data that they reference. These two concepts are intentionally disconnected since the data source can be used in many maps and projects simultaneously. If layer settings propagated to the data source, or vice versa, it would not be possible to use a single dataset in multiple maps or layers, without the configurations changing in all maps, every time someone made an edit to their individual maps.
If you are looking to edit the schema of the data itself, you can either open the fields view from the data source in the catalog, or, if you prefer entering the fields view through the layer in the TOC, you can configure the data source by selecting 'Data Source' in the 'Current Layer' drop down.