As long as you're on the same device, with the same IP address, you can use the same code to access the live stream.
Note that using Private browsing or Incognito mode in your web browser can cause issues if you reload the page and your IP address has changed. It is advised to not use this functionality within your web browser for a smoother experience.
If your IP address changes, as long as you are not using Private browsing or Incognito mode, and you have cookies enabled on your web browser, you should have a seamless experience should you need to reload the page or even come back later for on-demand viewing.