Unlike screenshots, the circled image you share is not saved locally on your device. Also, note that the resolution of the image you share will be relatively low.
If the clipped item has text, you also get the option to copy or translate it.
Google is also testing an option where Circle to Search can search using your whole screen instead of requiring you to draw or scribble on a particular area.
If the change is not yet live on your phone, ensure you are running the latest build of the Google app