The icon/button that you describe/illustrate merely triggers display of the Mail Shelf (the Shelf being a new feature of iPadOS multitasking environment). Tapping the icon summons the shelf view - which will show you a thumbnail image of all the running instances of the Mail App. Where multiple instances are running, a simple tap of a thumbnail will instantly switch to that view.
The iPadOS multitasking environment permits you to run multiple “instances” or views of the same App, such as Mail - with each instance, perhaps having a different open inbox.
For example, if you have three running instances/views of Mail, tapping the icon you describe will open the shelf with three thumbnails - each corresponding to each view within Mail - an individual view perhaps relating to a mailbox or draft Mail message. Simply tapping a thumbnail will switch directly to that running instance/view of the Mail App.
This article from Appleinsider has a useful explanation of this new feature along with many of the updates to multitasking:
https://appleinsider.com/articles/21/08/12/how-to-master-multitasking-on-ipad-and-ipad-pro-in-ipados-15
Another useful resource:
Use multitasking on your iPad - Apple Support
The iPad User Guide is available both online using a web browser (such as Safari), or as an Apple Books download:
iPad User Guide - Apple Support
https://books.apple.com/book/id1567104892
You should perhaps note that iPadOS15 multitasking, shelf and associated features cannot be disabled - but do not have to be actively used.