Minimise and close to system tray are different things, please don’t confuse the two. As an example, I don’t want or need Skype taking up space on my taskbar when I’m not actively using it but I do want it to run in the background so I know when I get a message, not only that but by having it minimised to system tray (or notification area, whatever you want to call it) I can see my online status and change it if needed.
This is not a Twitch only thing. As Barry pointed out there are many communication apps that work exactly the same way, and to name a couple more there is the Battle.net Launcher and the Steam client. They all have background functionality and don’t necessarily need to take up taskbar space when it’s preferable to just have them in the system tray, and common practice is to have that functionality tied in to the close button.