You can right click on the file type and click Properties and then on the General tab click Change next to Opens With.. and select ArcMap. This will not help since you cannot open this directly from Windows Explorer. You need to be navigating in ArcCatalog or the Add Data button in ArcMap.
I am not sure if you can successfully open the shapefile in ArcMap by double-clicking the file from Windows Explorer, even after changing the default Open-With option.
Shapefiles are not meant to be opened like that. You might like to add it from ArcCatalog, or Add Data option, or Drag-and-drop the shapefile from Windows Explorer to ArcMap.
The reason I asked was because if you have to upload to ArcGIS Online or need to zip and I have AutoCAD which cause me to confuse which one to select if you know what I mean.