We prove that with a (2+1)-dimensional Toda type system are associated algebraic skeletons which are (compatible assemblings) of particle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra structures for the system from algebraic skeletons of some particular choice for compatible associated absolute parallelisms. In particular, by a first choice of the absolute parallelism, we associate with the (2+1)-dimensional Toda type system a trix-coaxial Lie algebra structure made of two (compatible) base triadons constituting a 2-catena. Furthermore, by a second choice of the absolute parallelism, we associate a dyx-coaxial Lie algebra structure made of two (compatible) base dyons, as well as particle-like Lie algebra structures made of single 3-dyons. Some explicit examples of applications such as conservation laws related to special solutions, and an inverse spectral problem are worked out.