We present an extensive search in the literature and Gaia DR2 for visual co-moving binary companions to stars hosting exoplanets and brown dwarfs within 200 pc. We found 218 planet hosts out of 938 to be part of multiple-star systems, with 10 newly discovered binaries and 2 new tertiary stellar components. This represents an overall raw multiplicity rate of 23.2$pm$1.6% for hosts to exoplanets across all spectral types, with multi-planet systems found to have a lower duplicity frequency at the 2.2$sigma$ level. We found that more massive hosts are more often in binary configurations, and that planet-bearing stars in multiple systems are predominantly the most massive component of stellar binaries. Investigations of multiplicity as a function of planet mass and separation revealed that giant planets with masses >0.1 MJup are more frequently seen in stellar binaries than small sub-Jovian planets with a 3.6$sigma$ difference, a trend enhanced for the most massive (>7 MJup) short-period (<0.5 AU) planets and brown dwarf companions. Binarity was found to have no significant effect on the demographics of low-mass planets (<0.1 MJup) or warm and cool gas giants (>0.5 AU). While stellar companion mass appears to have no impact on planet properties, binary separation seems to be an important factor in the resulting structure of planetary systems. Stellar companions on separations <1000 AU can play a role in the formation or evolution of massive close-in planets, while planets in wider binaries show similar properties to planets orbiting single stars. Finally, numerous stellar companions on separations <1-3 arcsec likely remain undiscovered to this date. Continuous efforts to complete our knowledge of stellar multiplicity on separations of tens to hundreds of AU are essential to confirm the reported trends and further our understanding of the roles played by multiplicity on exoplanets.
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