Mail on iPad 15.1 - explanation and function of this icon

The icon that you illustrate 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” of the same App, such as Mail - with each instance, perhaps having a different open inbox.

For example, if you have three running instances of Mail, tapping the icon you describe will open the shelf with three thumbnails - each corresponding to each running instance of Mail. Simply tapping a thumbnail will switch directly to that running instance 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

The iPad User Guide is also a rich source of frequently overlooked information. The 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 are very welcome.