I'm Miles from the Social Care team at Shopify.
The avatar connecting in the Shopify checkout is always there when a customer logs in. When checking out through a Shopify checkout, a customer has the option to save their details for future purchases, and this happens across all Shopify stores. Based on this, the avatar cannot be removed, but a customer would only see their own.
I'm interested to hear your thoughts on why you'd like to remove this, though! Let me know by replying to the thread, as I can always pass on feedback on your behalf!
I also just spoke with a colleague, and found out that another reason you are seeing a profile picture for this customer is because they have an account with Gravatar! Gravatar allows users to upload profile pictures which will be displayed on many different sites around the web.
I agree with removing avatars from checkout.
They really dont add anything to the user experience. And the design of the default avatar doesnt match most store designs. If it was simpler, grey, and smaller, it wouldnt be so out of place.