This has probably been asked before and I just can't seem to find it. I have my twitch account linked and I have been affiliate for a while now, but I can't seem to find the option to add an icon next to my name for it like some others have.

Do you have a sub badge? I believe you need a sub badge for the option to appear

That is correct. It behaves the same way twitch partners work and it pulls your badge from the API every so often. So no badge on twitch, no icon here.