>Do Google Express customers have to provide their loyalty cards?
No. I order stuff from Target via Google Shopping Express multiple times a week without a loyalty card. I didn't even know Target had loyalty cards, in fact! Also if you're logged into your account you go go here and see what loyalty cards you can type in: https://www.google.com/shopping/express/#SettingsPlace:
The list includes Costco, REI, Staples, etc. but not Target.