It was already sad here that this way doesn't work at all. But to be sure in it I've done it again. Logged out everywhere, cleared all cookies and a cache, logged in. Doesn't work. Tab's limit appears.
The problem still exists! Still prompting me on the tab limit when I don't have spotify in any other tabs. I even tried doing the run spotify playing on your phone and log in to your web player and it works yesterday. However, it doesn't work today again. Get the bug fixed, spotify!
I kept having the same problem too and was able to resolve it using the Spotify mobile app. At the bottom of now playing, just select the webplayer that you want it to play on "Connect to a device". This solution have fixed the issue for me.