There are a number of interesting icons that could apply in various situations, such as the VS icon, the Azure or Office/Exchange logos, SQL, GitHub, etc as well as various generic icons like a database drum or a terminal icon.
Adding new icon assets and giving them a good name should be easy for Microsoft-owned properties. Some may need to be invented (eg, C# has no logo, where F# has a rather spiff one).
Care should be taken to include variants for various contrast situations.
Giving them real names instead of GUID names would make them easier to use without having to track down random images on the web, or something buried in the file system.
Proposed icon set:
Potential third party icons, if they agree?
Idle thought -- Some of the generic set could no doubt be found in common fonts. XAML allows buttons to be defined using symbol fonts -- perhaps that can be an alternative mechanism for defining icons? It also would allow, then, for defining font color. Worth a separate feature proposal, perhaps?
I'd like to do this in the future, but I'm wary of adding what is effectively an _API_ (user settings take dependency on these things, we can't change them lightly). In my head it's blocked behind #1564 because that would give users a good way to _discover_ those icons and set them as profile values and drag/drop in their own icons and all that jazz.
Maybe support base64 encoded image?
e.g.

is simply a line in config
data:image/png;base64,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
Most helpful comment
e.g.

is simply a line in config