I am not able to add fonts either... I can't find the "add more" + button. I am running the latest version of Illustrator on iPad.
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I don't see the sharing icon in the upper right of your screen. This makes me wonder, what kind of subscription do you have? Did you buy the app via the App Store? Or do you have a Creative Cloud subscription of some type?
I wonder if this could be the reason that "add fonts" isn't available.
For reference, here's what I see (below). I have a full subscription, and I see the share icon. I wonder if your subscription type maybe doesn't support adding your own fonts?