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I edited/clarified my comment before I saw your reply—apparently the restriction to three dimensions only apples to continuous-time systems, which is what I had in mind when I posted the question.
I'm not sure what the author had in mind exactly, but it's not accurate that at least 3D is required in continuous-time systems.
A common example in 2D is the chaotic billiard.
I am not familiar with the book and haven't studied strange attractors.
The physical motion of the billiards is two-dimensional, but wouldn’t the phase space be four-dimensional (since it tracks both position and momentum)?
I think the reason chaos needs at least three-dimensional phase space for continuous-time systems is that the orbits in phase space can’t intersect (which the paths of the billiards in real space obviously do).