A saved button is showing up in the Google app for some users, located in the hamburger menu just under the Recent button. This goes hand in hand with the save button that's showing up in custom tabs which have been opened from the Google app, which as you may have guessed, when tapped this will save (bookmark) whatever page you're on and keep it stored in the dedicated saved page. You can see what the saved page looks like in the first gallery image below, and if it's your first time opening it before anything has been saved you'll see a collection of descriptive cards telling you briefly about what the page is for and what you can save there.
This is reportedly showing up for some users who are on at least version 188.8.131.52 of the Google app beta, as well as Play Services version 11.7.30 beta, so make sure both of those are installed and you should see these new changes pop up. The saved button in custom tabs opened from the Google app will be replacing the share button that is currently there for most users.
Once you open the saved page, you'll see both saved lists and tags, with lists being a collection of any saved pages that you may have stored there. Initially, when opening up a custom tab from the Google app and then tapping the save button in the top right corner, you'll be asked to save it to a list. If you don't already have any lists then you'll be able to create a new one. For example, if you open a custom tab that leads to you an article or webpage about food or a food item, then once you hit the save button you can create a new list and title it "food," then save any future pages in this list which have to do with the same category. As noted in the first image below, the saved page will also store things like favorite places to go or favorite images. Since this seems to only be available with the beta versions of the Google app and Play Services from the version numbers listed above, there's no telling when Google might roll it out for users on the non-beta versions.