If I have two App Tabs with the same icon (from the same site/company, e.g. Wikipedia watchlists), I want to be able to see which is which without clicking them or hovering over them. An obvious way would be to assign custom icons to them.

You can do that now by add-ons. But I agree this would be indeed useful feature.

If this were possible, Firefox could come with custom icons, with images like envelopes and whatnot.

This isn't something that we want to support in the browser core. Definitely sounds like a good job for an add-on, though!