Hi 123-123-123,
Welcome to Apple Support Communities. I understand that you're not seeing an option to add images to Photos on your iPhone when you've done a Google Image search using Safari on your iPhone. I know how convenient it is to be able to save photos you've searched for; I'd like to help. I'd recommend starting with the steps below:
If the behavior continues, please provide the following details:
Does this happen with all images you try to save to Photos?
Are you able to save images sent to you in other apps, such as Messages?
When viewing the screen in your screenshot, if you tap "Share" do you see an option to save to Photos?
Best Regards.