In this paper, we investigate the coupled cubic-quintic nonlinear Schrodinger equations with variable coefficients, which describe the effects of quintic nonlinearity for the ultra-short optical pulse propagation in a non-Kerr medium, or in the twin-core nonlinear optical fiber or waveguide. Under certain constraints on the variable coefficients in such equations, mixed-type (bright-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation, and such vector-soliton solutions are only related to the delayed nonlinear response effect and nonlinearity. Through the graphic analysis, we find that the delayed nonlinear response effect and nonlinearity can both affect the vector-soliton amplitude, while the vector-soliton velocity merely depends on the delayed nonlinear response effect. With the choice on the variable coefficients representing the delayed nonlinear response effect and nonlinearity, interactions between the amplitude- and velocity-unchanging, amplitude-changing, velocity-changing and amplitude- and velocity-changing vector two solitons are obtained. We see that the interaction between the vector two solitons is elastic. We also find that the interaction period of the bound vector solitons decreases as the increase of the delayed nonlinear response effect or increases as the decrease of the delayed nonlinear response effect, but is independent of the nonlinearity.