There's also a chance Google image search is looking at the filename of the image to get a bit more context. Does the reverse image search of the cloaked image still work if you rename it something unrelated?
Here's the original image: https://imgur.com/a/Td4rhoy
And the cloaked: https://imgur.com/a/cPCiCZo
These were both saved as JPG with compression level 8/12. I searched for the cloaked crop (96.jpg) first this time: https://imgur.com/a/FSehQWO
And the original crop (10.jpg) next: https://imgur.com/a/yx4jF0B
This time, Google reverse image search did better at identifying the name of the singer in the cloaked image, instead of just giving the band name for the uncloaked.
Not super scientific since we don't really know what's going on behind-the-scenes with Google reverse image search, but it's certainly one adversary that doesn't seem to be easily fooled if there are other images of "you" out there for it to find. I also tried these small crops in Google Lens with less success (I got unrelated portraits for both images, cloaked or not).
Surveillance software that purports to accurately identify a person across multiple images is not just looking for the same content with some visually insignificant modifications. It's reading your facial structure, attaching your name to it, and searching for it in every image received. Fawkes is working to defeat that specific use case, not all fuzzy matchers in general.
P.S. If you have a human assailant running a reverse image search for photos of you, I think you're well past the point that something like this could be expected to help.
However, if someone is willing to go to that level of effort, the target probably needs to aim for something a little more forceful than tricking Facebook's autotagger.