See Shannon's original paper, "Communication in the presence of noise".[1] In that view, where the goal is to reproduce the original signal, noise and information are not distinguished, because they look the same on the channel. Both "increase the fractal dimension". Adding noise will make a lossless compression of something much larger. Go add some noise to a simple picture in Photoshop, then export in some lossless compressed format, like .png.

Now, lossy compression algorithms, such as JPEG, MP3, and almost all video compressors, are different. They have some model of what the content is "supposed to be like", and fit to that model. This works badly for content that does not fit the model, such as the mess JPEG makes of hard edges.

The original author referenced Shannon. The tests he's running use signals from which he cannot extract meaning, so he can only extract the statistics you can compute from a signal you do not understand. There are other tests for "is it signal or is it noise" - looking at the spectrum, autocorrelation to look for repetition (they did some of that), correlation with other signals, and such - but "fractal dimension" isn't one of them.

[1] https://course.ccs.neu.edu/csg250/ShannonNoise.pdf

Adding one random bit increases the entropy by a bit because there are now two possible messages - your message with a 0 or with a 1. Add a bunch of noise and you add a bunch of bits; entropy is average bits per message.