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We present the results of ALMA observations toward the low-mass Class-0 binary system, VLA 1623Aab in the Ophiuchus molecular cloud in $^{12}$CO, $^{13}$CO, and C$^{18}$O(2--1) lines. Our $^{12}$CO ($J$=2--1) data reveal that the VLA 1623 outflow consists of twin spatially overlapped outflows/jets. The redshifted northwestern jet exhibits the three cycles of wiggle with a spatial period of 1360$\pm$10 au, corresponding to a time period of 180 yr. The wiggle-like structure is also found in the position-velocity (PV) diagram, showing an amplitude in velocity of about 0.9 km s$^{-1}$. Both the period and the velocity amplitude of the wiggle are roughly consistent with those expected from the binary parameters, i.e., the orbital period (460$\pm$20 yr) and the Keplerian velocity (2.2 km s$^{-1}$). Our $^{13}$CO and C$^{18}$O images reveal the nature of the dense gas in the two cm/mm sources, VLA 1623-B and -W, and its relation to the outflows, and strongly support the previous interpretation that both are shocked cloudlets. The driving sources of the twin molecular outflows are, therefore, likely to be within the VLA 1623Aab binary. The axes of the two molecular outflows are estimated to be inclined by 70$\arcdeg$ from each other across the plane of sky, implying that the associated protostellar disks are also misaligned by $70\arcdeg$. Such a misalignment, together with a small binary separation of 34 au in the one of the youngest protobinary systems known, is difficult to explain by models of disk fragmentation in quiescent environments. Instead, other effects such as turbulence probably play roles in misaligning the disks.