Hoeg Law uses Suzy Lu cannot copyright her own face, and neither can anyone else copyright a face or the appearance of a living thing, because it was not created by human effort.
If you use a face to make a living, you might be able to trademark it. Or maybe in some context publicity rights (those two examples probably only your own likeness – you can’t also suggest someone else endorsed it, unless they did with a contact).
The Copyright Office has published a paper on Section512 of Section 17, regarding DMCA takedown safe harbor. Pages 36-38 do not seem that friendly to amateur users who are thought to have infringed on corporate media (there is an element of job protectionism maybe). The paper also makes comparison to the EU (with its Copyright Directive) and seems to think it does not miss the US notions of fair use and safe habor. This could be pushing the US more toward the EU model which has become hostile to amateur content. Electronic Frontier Foundation commented with a tweet today.
I note that the CASE Act still is not reported to have passed the Senate, given the distraction of COVID. So the Copyright Office can’t put forth its small claims court yet.