Local notifications on loginwiki, seen as cross-wiki notifications from loginwiki on other wikis, are generated by the LoginNotify extension for things like failed logins. However, on 2020-10-06 (and presumably beforehand?), these can't be marked as read.
Locally, it fails with:
{"error":{"code":"writeapidenied","info":"You're not allowed to edit this wiki through the API.","*":"…"},"servedby":"mw2306"}
Remotely, it fails with:
{"query":{"echomarkread":{"result":"success","errors":{"loginwiki":{"code":"writeapidenied","info":"You're not allowed to edit this wiki through the API.","*":"…"}},"alert":{"rawcount":1,"count":"1"},"message":{"rawcount":0,"count":"0"},"rawcount":1,"count":"1"}}}
Users indeed do not have the writeapi permission, and though git blame isn't helpful, it looks like it's been this way for a while. However, in T179765: Can't dismiss failed login attempt notification (or other notifications from login.wikimedia.org) this was noted as working, so presumably something changed.
The "easy" fix is to grant users the writeapi permission, but this might have security implications, so opening a ticket first.