If you are using Firefox instead, then head over to Mozilla Addons and look for App for Messenger™ there.
I tried both of those here and they do open up messenger in the browser upon clicking on their icons.
I'm a Firefox user. Did you try audio/video calls in Firefox by this method? I'd like to know this before I experiment with an add-on that has very few users and is not security-checked by Mozilla. I suspect that whatever issue prevents calling (audio or video) in a normal browser tab would not be solved by this or any of the other similar add-ons, and if that's true, I have no interest in an add-on just to change the interface.
The only reason I care about apps at all is that Meta no longer supports Firefox for calling. For years, I was perfectly happy to keep a tab open with messenger.com in it and do everything there. But some months ago Meta made a change that prevented calling in Firefox - when a call comes in, it will tell me to download the desktop app. I did install the app just to deal with calling, but the two don't talk to each other - if I have the tab open and a call comes in, it doesn't launch the app but instead continues to suggest that I download it! If neither the app nor a browser tab is open, I get no notifications of anything (if the app worked like Skype and would ring and open itself, that would be great, but it doesn't). So recently I have resorted to keeping the app open at all times (instead of a Firefox tab, since I don't need both), but then links open in Edge, as this thread's topic. Sigh...