I have two notification alerts stuck in the ECP for an few months now and I cannot seem to get rid of them.

The alerts are about an pst import and pst export request. Both alerts state that the in/export has been queued. However on both my servers the Get-MailboxExportRequest comes back empty (even with | Get-MailboxExportRequestStatistics added).

Worth mentioning is that both the export and import request where successful at the time. Also other alerts that have popped up since then (an SSL cert expiration alert) have disappeared once their respective issues had been solved.

Get-Notification indeed gives me both messages, but sadly there is no "Remove-Notification" command.

Does anyone have any idea how to get rid of these two messages?

Notifications live in the the arbitration mailbox and are supposed be cleared by at a max of 30 days. Try running the managed folder assistant against the arbitration mailbox to force a clean.

Get-Mailbox -Arbitration | Start-ManagedFolderAssistant

Nothing, no change.

Ran your command with -verbose to see if anything weird showed but it looks clean.