The gist is that if you factory reset your console you can no longer play online at all. You will receive error code 006-0303 whenever you try to play online. There's no fix, and will probably never be. Wii U is also affected. Pretendo will still work.
This also includes system transfers. I have a 2DS that I turned on after a system transfer months ago, and it receives error code 006-0303 because I did the setup process after this change was made. Not sure if both consoles are affected or just the source console, but I'm not going to test that out.
Edit: NNID doesn't matter. Factory resetting at all after this change means you can't play online anymore.
Pokemon Bank will not work on a factory reset 3DS.
Edit: It's been fixed. https://pretendo.network/blog/12-27-23