The first AMS-02 measurement confirms the existence of an excess in the cosmic-ray positron fraction previously reported by the PAMELA and Fermi-LAT experiments. If interpreted in terms of thermal dark matter (DM) annihilation, the AMS-02 result still suggests that the DM annihilation cross section in the present day should be significantly larger than that at freeze out. The Sommerfeld enhancement of DM annihilation cross section is a possible explanation, which is however subject to the constraints from DM thermal relic density, mainly due to the annihilation of DM particles into force-carrier particles introduced by the mechanism. We show that the effects of the Sommerfeld enhancement and the relic density constraints depend significantly on the nature of the force-carrier. Three scenarios where the force-carrier is a vector boson, scalar and pseudoscalar particle are investigated and compared. The results show that for the case with vector force-carrier, the Sommerfeld enhancement can marginally account for the AMS-02 data for DM particle annihilating into 2µ final states, while for scalar force-carrier the allowed Sommerfeld enhancement factor can be larger by a factor of two. For the case with a pesudoscalar force-carrier, the Sommerfeld enhancement factor can be very large in the resonance region, and it is possible to accommodate the AMS-02 and Fermi-LAT result for a variety of DM annihilation final states. *