The funky "gray window" with weird glyphs in place of text is Adwaita (GNOME and GTK's default theme), with no icon set other than the builtin icons in GTK (hence the sidebar has icons but not the buttons on the bottom of the dialog), without a font (hence the blocky "placeholder" glyphs).
This would tend to indicate to me that the app from the Snap is being blocked from reading at least those files pertaining to the font, the icon set and the theme.
Is the rest of the Discord client in disarray like that? Any clue if it's a Qt application? Interestingly Qt applications often can appear with GTK file dialogs but themselves are not themed to look like GTK. I'm out of guesses right now otherwise.
EDIT:
Ohhh! No wonder! Discord is an "app" based around Chrome!
AFAIK, Chrome doesn't follow your GTK theme (except for Open dialogs and stuff like that), the app seems to provide its own fonts (so there's no need for the external fonts GTK needs), and Chrome uses no GTK icons, so it's really no wonder that somebody could make a slip-up like that and forget to test the minor GTK-using part of the app. But the idea of a slip-up is just speculation on my part.