Shopping list is run through Google Express. If you add an item through Google Assistant it adds it to the shopping list in Express. If you do not have the app downloaded it adds it through the ge website.
It's dumb. If you add anything to any list, it puts it on my shopping list. "add hiking to my weekend list" it puts hiking on my shopping list.
I have found no voice controlled shopping list for my phone other than Google Express not even Bixby through Samsung notes.