What would produce this "hamburger" corruption on images?
JPEG projects $8\times 8$ blocks of images onto $64$ 2D cosine patterns:



The one in column $1$ and row $5$, once quantized, may look like your hamburger. Luminance and chroma components may get different subsampling patterns. I suspect that the low varying background is nearly horizontal, and due to the different processing steps, it ends up with a mid-frequency non-zero component. My main issue is its $10\times 10$ apparent size. Apparently, WebP also does some prediction on blocks.

You might be interested in compression forencics, that try to recover the history of compression. For JPEG, JPEG CHEst (JPEG Compression History Estimation for Color Images) does the job, based on paper JPEG Compression History Estimation
for Color Images.