Favicon for media files

I used this site to create all the necessary icons: https://realfavicongenerator.net/. It can start with a .png file.

I'll add a couple more thoughts. First, I chose this site because it creates icons customized for all platforms. That may be overkill for you. Second, it's inherently risky to use an online tool to create files that you load to your site. I manually checked the files this site generated to be sure they looked clean, but please use caution if you use this or any online tool.

Ok so I can get the favicon to show up in the browser as an icon, but when I send the my web link out as a text (iphone), the favicon doesn't show.

I don't believe the favicon is supposed to show when you share the link as a text.

You might need to use a plugin like YoastSEO to set a specific image for that.