While you have unread direct messages, or unread messages in one of your groups or in Magpie support, each open Magpie tab shows a dot on its icon and the number of those conversations in front of its title, such as (2) Chat. Team channels are not counted. The mark clears once you have opened those conversations. Safari may keep the tab's icon without the dot, but it shows the number.
To turn it off, open Account settings and switch off Unread messages on the browser tab. The Message sound section of Account settings is now called New messages.