Do you also happen to have your PSN ID connected to Discord? If so, that'd show it on your profile, too.
Check your Discord Connections in your Discord Settings; if you have your PSN ID connected, it should show an option to hide it, like having an XBL ID linked does.
As far as how Discord displays your connected accounts, that is entirely Discord's system. Bungie cannot assist you with Discord's system, and Discord stating that Bungie is making it display that and Bungie has to fix their system so it doesn't seems a bit odd to me.
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I have linked it, but I have it hidden similarly to how, like you said, you can hide your Xbox name, or hide any name. Going further, I then even completely unlinked it and it still showed my PSN ID.
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As a workaround, you could just add your BungieName to your status. I'd also suggest that you give feedback to Discord support to contact Bungie concerning what might be making their system not display the information they're trying to make it display. Based on what I've seen other websites do, and how Charlemagne works, the Bungie.net API does seem to allow pulling the BungieName, so it's odd that Discord would have it coded to display your active save data's platform account instead of your BungieName.