There is an increasing demand for large format detector arrays with large bandwidths and high antenna efficiencies for future THz astronomical radiometric applications. For direct detection instruments, it is also desired to have antennas with dual polarization reception in order to increase the received power from incoherent sources, thereby improving the observing speed of the instrument. The main goal of this work is the validation of the incoherent detection of two orthogonal polarizations by a leaky lens antenna, coupled to a single Microwave Kinetic Inductance Detector (MKID). Depending on the absorbed power over a distributed transmission line, the resonant frequency of the MKID changes. The proposed antenna is composed of two crossed leaky wave slots feeding a silicon extended hemispherical lens. The slots are coupled to four aluminum (Al) coplanar waveguide (CPW) lines that incoherently absorb the incoming THz radiation. The antenna and the power absorbing CPW lines are embedded inside the MKID, allowing an efficient radiation detection at THz frequencies where no lossless superconductors are available. The proposed dual-polarized device absorbs power incrementally over four different CPWs incoherently and is therefore simulated in reception (deriving a plane-wave response) similarly to what is done in distributed absorbers. We compare numerically and experimentally the proposed dual-polarized leaky lens coupled MKID and its single polarization counterpart and show that the dual polarized device receives twice as much power as the single-polarized one. Eventually, the dual-polarized device, when used with air-bridges, provides the same angular selectivity and twice the throughput of the single-polarized one.