Logo or Icon - which is used when site is shared

I have just published a site for someone who had provided a site icon and a site logo. The logo has a tagline, the icon doesn’t. The client has just shared their shiny new site asked why is the text missing, so I’ve explained it is the icon that gets shared, not the larger logo file.
Furthermore, as the icon is also the favicon I don’t think adding the text is an option to such a low res file. Just wondering is there a way to instruct a site to use the logo instead of icon when sharing, or is that dependent upon the device doing the sharing?