Red blood cell (RBC) aggregates play an important role in determining blood rheology. RBCs in plasma or polymer solution interact attractively to form various shapes of RBC doublets, where the attractive interactions can be varied by changing the solution conditions. A systematic numerical study on RBC doublet formation is performed, which takes into account the shear elasticity of the RBC membrane due to the spectrin cytoskeleton, in addition to the membrane bending rigidity. RBC membranes are modeled by two-dimensional triangular networks of linked vertices, which represent three-dimensional cell shapes. The phase space of RBC doublet shapes in a wide range of adhesion strengths, reduced volumes, and shear elasticities is obtained. The shear elasticity of the RBC membrane changes the doublet phases significantly. Experimental images of RBC doublets in different solutions show similar configurations. Furthermore, we show that rouleau formation is affected by the doublet structure.