We present a continuous microwave photon counter based on superconducting cavity-coupled semiconductor quantum dots. The device utilizes photon-assisted tunneling in a double quantum dot with tunneling events being probed by a third dot. Our device detects both single and multiple-photon absorption events independently, thanks to the energy tunability of a two-level double-dot absorber. We show that the photon-assisted tunnel rates serve as the measure of the cavity photon state in line with the P(E) theory-a theoretical framework delineating the mediation of the cavity photon field via a two-level absorber. We further describe the single photon detection using the Jaynes-Cummings input-output theory and show that it agrees with the P(E) theory predictions at a low-power regime.