We report on a spectroscopic program to search for dual quasars using Subaru Hyper Suprime-Cam (HSC) images of SDSS quasars, which represent an important stage during galaxy mergers. Using Subaru/FOCAS and Gemini-N/GMOS, we identify three new physically associated quasar pairs having projected separations less than 20 kpc, out of 26 observed candidates. These include the discovery of the highest-redshift (z = 3.1) quasar pair with a separation <10 kpc. Based on the sample acquired to date, the success rate of identifying physically associated dual quasars is 19% when excluding stars based on their HSC colors. Using the full sample of six spectroscopically confirmed dual quasars, including three previously published, we find that the black holes in these systems have black hole masses (M BH ~ 108-9 M ⊙), bolometric luminosities (log L bol ~ 44.5-47.5 erg s-1) and Eddington ratios (0.01-0.3) similar to single SDSS quasars. We measure the stellar mass of their host galaxies based on 2D image decomposition of the five-band (grizy) optical emission and assess the mass relation between supermassive black holes (SMBHs) and their hosts. Dual SMBHs appear to have elevated masses relative to their host galaxies. Thus, mergers may not necessarily align such systems onto the local mass relation, as suggested by the Horizon-AGN simulation. This study suggests that dual luminous quasars are triggered by mergers prior to the final coalescence of the two SMBHs, resulting in early mass growth of the black holes relative to their host galaxies.