We present a detailed characterization of 2MASS J04435750+3723031, a low-mass companion orbiting the young M2 star 2MASS J04435686+3723033 at 7.″6 (550 au) with potential membership in the 23 Myr β Pictoris moving group (βPMG). Using near-infrared (NIR) spectroscopy of the companion from IRTF/SpeX, we have found a spectral type of M6 ± 1 and indications of youth through age-sensitive absorption lines and a low surface gravity index (VL-G). A young age is supported by Hα emission and lithium absorption in the host. We reevaluate the membership of this system and find that it is a marginally consistent kinematic match to the βPMG using Gaia parallaxes and new radial velocities for the host and companion. If this system does belong to the βPMG, it would be a kinematic outlier and the companion would be overluminous compared to other similar ultracool objects like PZ Tel B; this would suggest that 2M0443+3723 B could be a close binary (≈52+52 M
Jup if equal-flux, compared with 99 ± 5 M
Jup if single), and would make it the sixth substellar companion in this group. To test this hypothesis, we acquired NIR adaptive optics images with Keck II/NIRC2, but they do not resolve the companion to be a binary down to the diffraction limit of ∼3 au. If 2M0443+3723 AB does not belong to any moving group, then its age is more uncertain. In this case it is still young (≲30 Myr), and the implied mass of the companion would be between ∼30 and 110 M
Jup.