There is a visual resemblence between zoom levels because of the statistical self-similarity across zoom levels. I suppose the word resemblence was chosen by the author as a less strict variation of the concept. But even the self-similarity of for instance the convergence border of the Mandelbrot set isn't completely self-same either.

Yeah if you were to discretize it, then you'd get random walks with the only difference being the variance of movement from the last point. Wish I could find this one source that talked about continuous time noise processes - it was pretty interesting. If you assign a random variable to every point t on a continuum, it's not a real process (in terms of the definitions set forth in Measure Theory), so you have to use other methods to define them, like limits of discrete processes.

You should read Mandelbrot’s book “the fractal geometry of nature.” He talks quite a bit about ideas like this. It’s written with the intention that non mathematicians can read and understand the ideas too which is kind of neat.

Well yeah, that's the "(statistically) selfsimilar" bit. Random fractals like this aren't self-congruent, just self-resembling.