Is it possible to make a button which, when tapped, opens an overlay which displays information (not an external app)

I would like a button/keyboard shortcut where, when I tap it, will open a little overlay window (similar to the BTT Control Center preset) that shows information like world times in cities I choose, the weather for the next few days, etc.

How can I do this?

You can do this using the "Floating Webview/HTML menu" action. There you can use custom HTML to do whatever you want.