In this work, we study Riemann problems and delta-shock solutions for a nonsymmetric Keyfitz-Kranzer system with a Coulomb-like friction term or linear damping. We show the existence of an intricate delta-shock wave solution and its generalized Rankine-Hugoniot condition resulting from the analysis. In particular, we also show the existence of a shock wave solution satisfying the classical Rankine-Hugoniot condition and the Lax shock condition, which is supported by the corresponding homogeneous Keyfitz-Kranzer system under investigation. Some numerical results exhibiting the formation process of delta-shocks are also presented, verifying the theory being presented. In particular, the robustness of the numerics is illustrated with a very interesting linear damping example, where we show a simulation of the cutoff time in which a delta-shock singular solution ceases to exist, and in fully agreement with the theoretical results.