The author here. Emergent algebras and dilation structures appeared from the effort to find an answer to a problem by Gromov in sub-riemannian (aka Carnot-Caratheodory) geometry: "Develop a sufficiently rich and robust internal CC language which would enable us to capture the essential external characteristics of our CC spaces". Here is a short description with links of the problem.
This problem is to recover the differential structure from the metric structure, which is a matter of analysis more than one of geometry. In the particular case of riemannian geometry this was solved by Nikolaev in 1998-1999.
A solution to this problem is in this article, which uses dilation structures.
Another application is more algebraic, namely to characterize what could be the correspondent of affine geometry for Carnot groups, a particular class of nilpotent groups which generalize vector spaces (and they appear naturally in SR geometry, but also elsewhere, for example the Heisenberg groups are the simplest non-commutative Carnot groups).
Yet another application, which is work in progress, is to treat the formalism of emergent algebras as yet another interaction graphs theory.