Is it possible to use the Maps app to save a location and give it a name so that all the saved locations appear as a layer? This seems like a option that should exist in the Maps location app but I cannot seem to find it. 

Backstory: My previous phone was a Windows Phone. On this phone the default location app HERE Maps showed maps. Long pressing an area on the map area gives an address which if tapped would allow to name the place and mark it as favorite. All the favorites would show up on the map above the base layer.

Now: The Maps app with the android phone has a similar option. Long pressing the map area will give the address of an area. Tapping the address allows you to mark it as a favorite. And all the favorites show up on the map. However it is missing the option to name a place to show up on the map. Is there anyway this can be done with the Maps app?
It's possible to create a personal layer in Google Maps and have it show up in the Maps app. You can create a pin, and label it. However, you will only see pins on the app. But once you click on the pin, you will see your custom text.

Google Maps site. Log-in to Google Maps in Desktop mode (a PC maybe?). At the top left beside the Get directions button, click the My Places button. Then, click the red Create Map button. Give your map a title, add some pins (and custom text) on it, then click the Done button.

Google Maps app. If the app is open, close it and open it again. Once open, click the Layers button at the bottom right. Click My Maps and your list of custom maps will show up, together with the pins you put up. 

See if that's a suitable workaround.

Update: The My Maps option of Google Maps has been discontinued. As of 12th December 2013, a new app, the Google Maps Engine android app has features similar to My Maps option that allows you to create maps from the Google Maps Engine site. This app lets you see it the created maps and layers.