The shift-invariance operator
The central notion is that  the considered algebras are generated by a shift operator $\mathcal{S}$. Every element (filter) of the algebra should be (by definition) generated by a (finite) number of algebraic operations (addition, product, scalar multiplication) involving the generator and its inverse. This is explained in more details in Section II-A, around Equation (30), and footnote 19. So if the shift commutes with any filter, since any filter is made of a combination of shifts and inverse shifts, then any filter commutes with any filter.
To me, this is really related to the subset of shift generated algebras they consider (and not on the module of signals).
For instance, with standard discrete filters this unit element would be the Kronecker $\delta_0$ (the &quot;unit&quot; algebra element relative to the multiplicative $\cdot$ operation of the algebra, here the convolution). A (finite support) filter would be written as sums and multiplies of shifts, with scalars corresponding to filter taps: $$h_0S^0(\delta_0)+h_1S(\delta_0)+h_2S^2(\delta_0)+\cdots$$
and commute with basic shifts.