Welcome to iMore! The app switcher is built into iOS and cannot be disabled. It's the only way to close problem apps without resetting your phone. I'm not sure how you're seeing messages in the app switcher unless you're just speaking about the thumbnail that shows from the Messages app. Again, that feature cannot be disabled.
If there is a particular app that you don't want to show in the app switcher, you can always swipe up and remove it. Keep in mind though, if you open the app back up it will show up again.