Google Assistant may have a Screen Search button coming as a new thread on Reddit contains the discovery of one individual explaining that when viewing something on their phone, say a particular page within the Reddit app, and after pressing on the home button to bring up Google Assistant, they're now seeing an option that asks "what's on my screen?" sitting next to the option to share what's on their screen at the current time through a screenshot. This appears to be a feature that Google has in testing on a select number of devices as it's not showing up for everyone who has Google Assistant on their device.
Since it's in testing, there's it seems that Google hasn't shared any such information on this particular new change just yet, and there's no information on when they might decide to roll it out to users on a wider scale, though there is always the possibility that they could be planning this for a release to the public in the near future, and right around the time they send out the public update for Android 7.1.2 Nougat could be the perfect time for them to do so.
This implementation doesn't appear to be a time saving measure, so it's not entirely clear why Google would have it incorporated instead of having Google Assistant just automatically search for what's on screen when it's opened up on top of whatever a user is already doing. For those that preferred Now on Tap to the way things are now, there's a method to switch back to it from Google Assistant even on Pixel devices and the process isn't all that difficult to complete, though there's no point in attempting it if you don't care too much about the changes that have been made. On a somewhat unrelated note, some of the other users in the same Reddit thread where this new potential screen search button was discovered mention that they want Google to bring back the text selection option for when Google Assistant is brought up, which used to be an available option with Now on Tap. While there is no indication that Google will re-implement this, it seems that it's something users do want to have.