We present SCExAO/CHARIS high-contrast imaging/$JHK$ integral field spectroscopy of $kappa$ And b, a directly-imaged low-mass companion orbiting a nearby B9V star. We detect $kappa$ And b at a high signal-to-noise and extract high precision spectrophotometry using a new forward-modeling algorithm for (A-)LOCI complementary to KLIP-FM developed by Pueyo (2016). $kappa$ And bs spectrum best resembles that of a low-gravity L0--L1 dwarf (L0--L1$gamma$). Its spectrum and luminosity are very well matched by 2MASSJ0141-4633 and several other 12.5--15 $M_{rm J}$ free floating members of the 40 $Myr$-old Tuc-Hor Association, consistent with a system age derived from recent interferometric results for the primary, a companion mass at/near the deuterium-burning limit (13$^{+12}_{-2}$ M$_{rm J}$), and a companion-to-primary mass ratio characteristic of other directly-imaged planets ($q$ $sim$ 0.005$^{+0.005}_{-0.001}$). We did not unambiguously identify additional, more closely-orbiting companions brighter and more massive than $kappa$ And b down to $rho$ $sim$ 0.3 (15 au). SCExAO/CHARIS and complementary Keck/NIRC2 astrometric points reveal clockwise orbital motion. Modeling points towards a likely eccentric orbit: a subset of acceptable orbits include those that are aligned with the stars rotation axis. However, $kappa$ And bs semimajor axis is plausibly larger than 75 au and in a region where disk instability could form massive companions. Deeper $kappa$ And high-contrast imaging and low-resolution spectroscopy from extreme AO systems like SCExAO/CHARIS and higher resolution spectroscopy from Keck/OSIRIS or, later, IRIS on the Thirty Meter Telescope could help clarify $kappa$ And bs chemistry and whether its spectrum provides an insight into its formation environment.
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