I did not happen to attempt to control that aspect, but when using
https://realfavicongenerator.net/ to generate cross-platform image assets for a favicon image, after uploading the image and you're being shown the different images and options you have for iOS, Android, Windows Metro and Apple Safari, down at the bottom of that page in the "Favicon Generator Options" section, there is an "App Name" tab that lets you override the name of your site.
It doesn't appear that this uses specifically the Open Graph metadata in order to achieve the result; it appears to generate
<meta name="apple-mobile-web-app-title" content="My Site Name">
and
<meta name="application-name" content="My Site Name">
. But it might still achieve the end result you are desiring on the device, and might even do it in a broader set of scenarios instead of "only those that support Open Graph."
In a phpBB context, after uploading the
https://realfavicongenerator.net/ generated image assets and other files to your site, you would typically edit the overall_header.html in the phpBB style you're using, to include the recommended metadata and references to those image assets. You could always add the Open Graph metadata there too, just for additional good measure.
I would assume such a link is otherwise showing the <title> description, which on phpBB is dynamically generated to show pending notification information, and page name in addition to site name. Which yes, might not be exactly what you would choose to be "a name or description of your site."