In this demo, I take you through the following:
adding and checking items on the Shopping List using Google Home,
Adding items from the Google Assistant app running on my iPhone, and
deleting, editing and adding items to the Shopping list using http://shoppinglist.google.com
In all cases for Google Home, Google Assistant, and the Shopping List website, I had previously logged in to my Google Account.
Just a final note, you'll need to download the Google Assistant app from the iOS App Store to use it on the iPhone: