Shapefiles are not capable of storing additional information, e.g. where to put labels, wich font and size and so on. Shapefiles only store geometries (like the points in your example) together with attribute information.
The information you wish to store can be saved in a qgis layer definition file (*.qlr) or in a qgis project, rsp.
The layerfile will store WHERE your shapefile is (the data source) and HOW it is displayed (and some further settings like editors and so on). You can achieve this by right clicking on the layer (
NC_with_answer  in your example) and then select 'Save layer definition...' (hint: I unfortunaltly have the german GUI and hope to translate this correctly).
The layer definition file can be re-opened by the Layers-Menu > Add from layer definition...
Here is a technical description of what shapefiles are capable of: https://www.esri.com/library/whitepapers/pdfs/shapefile.pdf and I strongly suggest reading the documentation at qgis.org!