Can SPO support multiple stores on the same browser?
No, unfortunately. There is a cache issue related to session storage. When you use the same browser to access different stores and SPO, the session cache will mess up. It's NOT related to logging on and out on Shopify. Also, uninstalling and reinstalling SPO will NOT solve the issue.
For the time being, you can use a Chrome incognito tab or another browser to access SPO, the session cache will be stored separately. In the long run, we'll try to see how we can improve the issue.