According to your post that says
You might want to establish a browser session once you authenticate. That way when any calls come to your App, you just check if a session exists. If not, go through the login task and establish one, then carry on with the request. The only other time you need to validate with those parameters is when requests are coming from Shopify that are Webhooks or App Proxy calls, since those clearly fall outside the realm of a session.
Once you master that, you can play with the idea of one browser session but two or more shops connecting to your App. That is an added level of tricky.
If you are not following that pattern you are off on a weird experiment of your own, and it is unlikely anyone will help you in your endeavors.
But my problem start with session as I explained below-
I am writing a shopify app to administer a store. Our app uses Embedded App SDK. The problem is that if a user has more than one shops, he cannot open two instances of our app to manage two different stores in the same browser window since I store the current app in user's session. Now these two tabs will show the same content.
Could you give me some direction about how can we implement our app so that user can open two browser tabs and each tab corresponds to the correct app?