How to disable small tile responses in the Messages and Mail app on iPad?

You have encountered the Shelf - this being an integral element of iPadOS multitasking.

iPadOS permits you to have multiple App windows of the same App - each being self contained. The shelf, when triggered, shows a thumbnail of each Window associated with the App. Each App - such as Mail, Safari, Notes, Calendar, etc. - has its own corresponding shelf.

The Shelf is used to quickly switch between open windows or instances of the App. Individual windows can be closed directly from the Shelf; just swipe the unwanted thumbnail off the top of the screen.

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 online using a web browser:

iPad User Guide - Apple Support

The feature you refer to is called a “Shelf”. Read about it here.

Use multitasking on your iPad - Apple Support