Website selector could use an option to display URLs instead of names. Or names along with URLs.
Adding the URL to the displayed value or the tooltip (title attribute) is quite easy. Making it switchable is a bit more complex I guess as we need to adjust the autocomplete, search etc.
This should be developed in a new - custom - plugin. This would cause too much pain as a setting in core over time. We could possibly add an attribute like
<a ...>sitename</a> so that another plugin could listen to this attribute and replace the site name with the link from
href. This should work. The site and autocomplete does likely not need to be adjusted as this is happening server side and already searching in name, url, ... but I'm not 100% sure.
One more reason why I don't want to have this in core: Future "sites" or "measurables" won't have URLs. Eg Mobile websites. Therefore we need to always display the name. This is a custom feature request that has to be developed in a custom plugin.