See also: Where do EDITOR, PAGER, BROWSER environment variables come from?
EDITOR and PAGER are mentioned in the standards as belonging to variables you'd be unwise to conflict with since they are widely used [...] Various programs respect various combinations of them [...]
The BROWSER variable is not in the same league as EDITOR or PAGER - it is not mentioned by the standards. However, some programs may use them.
IMO it could be best to consider variables for the GUI, as not being commonly used, unless you have a specific use case in mind. The reason is that the user will have set their preferences in their graphical desktop through the MIME system instead.
For example, Gnome Settings app lets you choose a browser, but it does not set the $BROWSER environment variable.
In this case you could try using
(xdg-open apparently has some special-cases e.g. for falling back to BROWSER. https://utcc.utoronto.ca/~cks/space/blog/linux/XdgOpenWhichBrowser But that probably means if you want a browser, you might as well use xdg-open and let it handle things for you).