Acidizing is a commonly used stimulation technique in the development of carbonate reservoirs. Besides matrix carbonate rocks, fracture-vuggy carbonate reservoirs are also targets for acidizing because the contribution of unconnected, complex media to fluid flow is limited. In this work, the two-scale continuum model is extended with the capability to study acid stimulation in fracture-vuggy carbonate rocks. The Darcy equation is employed to describe the fluid flow within the matrix and fractures, while flow within vuggs is described by the Navier-Stokes equations. The extended Beavers-Joseph-Saffman interface conditions are used to couple the seepage and free flow. In addition, the Biot elasticity model is introduced to coupling geomechanical process accounting for matrix elastic deformation, fracture normal closure, and vug deformation. Based on the extended model, sensitive cases are designed to the impact of stress on the dissolution patterns and acid breakthrough volume during the acidizing process in different fracture-vug combinations. Results show that the presence of fractures and vugs do affact and induce the movement of acid fluids within an acidizing system. The role of stress in acidizing is particularly pronounced in the presence of fractures. Stress affects the closure of fractures, with high stress reducing their inducing effect on the acid solution. However, the presence of fractures and vugs does not disrupt the equilibrium conditions for wormhole formation, meaning that it does not change the dissolution pattern corresponding to the specific injection rate.