I have made a polygon in Google Earth, but somehow there is no icon or feature for saving .shp. Is there any way to save a shapefile in Google Earth?
There is no export to shapefile via Google Earth.
These steps when using the Google Earth web version:
Once you are in the project section, you can click on the three vertical dots and then export as KML file as mentioned on this thread
These steps when using Earth Pro desktop version:
You will need to right-click the polygon layer, then click on Save As and select KML. Which you can then view in a GIS application/service. If needed, you can find a tool to convert your KML to shapefile.
whyzar is right that Google Earth does not have a way to save to a shapefile, but can save to a KML which can then be converted to a shapefile with other software. Since it looks like you're using the new web version of Google Earth (not the Earth Pro desktop version), the UI is a bit different than described. You'll want to use the 3-dot menu at the top of your project panel, where you should find an option to save as KML.