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We place observational constraints on a dark energy (DE) model in which aquintessence scalar field $\phi$ is coupled to dark matter (DM) throughmomentum and energy exchanges. The momentum transfer is weighed by aninteraction between the field derivative and DM four velocity with a couplingconstant $\beta$, whereas the energy exchange is characterized by anexponential scalar-field coupling to the DM density with a coupling constant$Q$. A positive coupling $\beta$ leads to the suppression for the growth of DMdensity perturbations at low redshifts, whose property offers a possibility forresolving the $\sigma_8$ tension problem. A negative coupling $Q$ gives rise toa $\phi$-matter-dominated epoch, whose presence can reduce the sound horizonaround the Cosmic Microwave Background (CMB) decoupling epoch. Using the dataof Planck 2018, 12-th Sloan Digital Sky Survey, Phantheon supernovae samples,and 1-year dark energy survey, we find that the two couplings are constrainedto be $\beta=0.417^{+1.592}_{-0.307}$ and $Q =-0.036^{+0.036}_{-0.010}$ at68\,\% confidence level (CL). Thus, there is an interesting observationalsignature of the momentum exchange ($\beta \neq 0$) between DE and DM, with apeak of the probability distribution of the energy transfer coupling at $Q<0$.

Observational constraints on interactions between dark energy and dark matter with momentum and energy transfers