Pairwise hydrodynamic interactions of microswimmers form the fundamental building blocks for understanding their more complex collective behaviors. In this work, we revisit the canonical problem of two interacting squirmers swimming along their common line of centers in both Newtonian and shear-thinning fluids. For the Newtonian case, we first derive an exact, closed-form solution for the axisymmetric Stokes flow generated by the interacting pair, thereby complementing prior analyses based on the reciprocal theorem approach by providing direct access to the detailed knowledge of the flow around the swimmers. The analytical solution is then used to cross-validate numerical simulations based on the finite element method. The combined theoretical and numerical investigation reveals co-swimming configurations in which the two squirmers develop identical velocities over a range of separations. We rationalize these behaviors through symmetry arguments and quantify their propulsion performance in terms of the speed and energetic cost of swimming. Furthermore, motivated by the prevalence of shear-thinning biological fluids encountered by microswimmers, we examine how this ubiquitous non-Newtonian rheological behavior modifies the propulsion characteristics of these co-swimming pairs. Taken together, our results provide a systematic characterization of an idealized model of pairwise squirmer interactions in Newtonian and shear-thinning fluids, establishing quantitative benchmarks for future studies of more complex and realistic swimmer configurations.