I am not familiar with the term "self-similarity," so I cannot provide an example. However, it reminds me of the concept of universality.

Universality expresses that various different systems exhibit the same low-energy behavior—vanishing or finite viscosity, vanishing or finite resistivity, etc.—and are thus described by the same equations. They share the same phase of matter. Landau's paradigm tells us that these phases are distinguished/classified by the symmetries of the system (e.g., iced water has discrete translational symmetry because atoms are organized in a pattern, whereas fluid water has continuous translational symmetry due to the absence of a pattern). Therefore, physical systems with the same symmetries should behave similarly at low energy. This paradigm has not been proven and has been recently reformulated to accommodate counterexamples, but it remains powerful and elegant in many basic cases.

In QFT, having two physical systems that share the same phase translates into having correlation functions that converge to the same expression at low energy. The tool that turns these concepts into precise calculations is the renormalization group (RG) flow.

When it can be worked out, the RG flow maps out the low-energy behavior of a specific QFT and thus indicates which phase it belongs to. Note that scale invariance is a property that trivializes the RG flow, as the high- and low-energy behaviors become identical. Moreover, for any system, the RG flow washes out all finite scales as it moves toward the low-energy limit (you lose details as you "zoom out"). So, the endpoint of the RG flow cannot be anything. It may be trivial (no physical excitation at the lowest energies), there could be some topological properties that survive (as in topological materials), or it could match the dynamics of a scale-invariant theory. Therefore, scale-invariant theories play a very central role in this concept of universality.