it has a notification history feature (at least on PC), it's just impossible to find and deleted messages disappear from it. Upper-right corner is your "inbox" which is totally worthless, and tabbed behind that is your notification history. I use it to find totally buried @Mods pings that I missed by thousands of messages and that's about it, it's not good.
You can usually get most of your notifications with Ctrl+[T or K] and then going through the menu here, but for reasons I've yet to figure out sometimes DMs don't show up here even when they have unread messages. I think there is some incorrect logic that kicks in when you have a high number of unread channels and it can't show as many "previous channels" as it wants to.
None of this is to defend Discord, I think their UI is bad and I've hated the way DMs function since day 1, and every single part of their app that relies on frecency (or doesn't but should) is abysmal (reaction autocomplete, the reaction pop-up menu, ctrl+T when you start typing something, the mention autocomplete behavior in any channel)
But once you learn the poorly-documented navigation flow of ctrl+T and then using @, #, or * to filter users/channels/servers it gets easier to use. My current biggest complaint is lack of a "previous channel within this server" hotkey, "previous channel that you visited globally" exists but you can't restrict it within one server, and it makes navigating some of my servers an absolute nightmare.
I get on Discord here and there to discuss Linux, programming and so on. It is definitely much different than IRC. The demographic seems to be 14 year olds trying to customize their WM.