How can I add an extension to my share menu in MacBook Pro?

No, did nothing here I'm aware of, I find Telegram.app in my Applications folder, last modified Aug 03,2019... I often download stuff just to see if I can help out here or elsewhere, I don't remember anything about it but either I did or it comes with something else.

Maybe try the non-Appstore download here...

https://macos.telegram.org/

Actually I have Whatsapp and Telegram installed on my Mac but don't see them in the share menu as you can see in the attached pictures. How did you add Telegram?

You missed the second part:

vendor has provided a sharing extension

If the developer didn't create an extension which they bundled with the app, then it won't show up even with the app installed.

Hmmm, Telegram is 5th from the bottom here...

Check System Preferences>Extensions, at least Telegram is available on mine.

Unless the software vendor has provided a sharing extension that registers after product installation, then these extensions can only be modified by Apple.

oops...