Is it possible to implement an extension that will be displayed on top of the site?

Question

Browser extensions

I want to make one utility for Chrome/Safari, which, when you click on the extension button on top of the side of the site, will show the necessary information. But so that it is not a popup, as they write in the guides, and does not close when you click on any other buttons / tabs, but that this panel relates directly to this tab.
Tell me which direction to go and what to read? And I don't quite understand how to do this.

Answer 1

They already answered about extensions: content scripts solve it there.
I will only add that the task at hand is much easier and more convenient to solve (without creating a separate extension) entirely on user scripts (GreaseMonkey/TamperMonkey). This is how I modify almost all (significant for me) sites with a file for myself. In addition to the convenience of development, it is also a plus that when distributing ready-made userscripts, their potential users immediately see during installation: 1) limited rights that the script requests; 2) a limited circle of domains (or any addresses) on which the script will be used; 3) you can view the source of the script during installation (in a very convenient form, not climbing on githubs). This is all very important for those who are worried about their security and do not want to let extensions from unknown authors with unlimited rights into their browser.