  The orbital eccentricities of directly imaged exoplanets and brown dwarf
companions provide clues about their formation and dynamical histories. We
combine new high-contrast imaging observations of substellar companions
obtained primarily with Keck/NIRC2 together with astrometry from the literature
to test for differences in the population-level eccentricity distributions of
27 long-period giant planets and brown dwarf companions between 5-100 AU using
hierarchical Bayesian modeling. Orbit fits are performed in a uniform manner
for companions with short orbital arcs; this typically results in broad
constraints for individual eccentricity distributions, but together as an
ensemble these systems provide valuable insight into their collective
underlying orbital patterns. The shape of the eccentricity distribution
function for our full sample of substellar companions is approximately flat
from e=0-1. When subdivided by companion mass and mass ratio, the underlying
distributions for giant planets and brown dwarfs show significant differences.
Low mass ratio companions preferentially have low eccentricities, similar to
the orbital properties of warm Jupiters found with radial velocities and
transits. We interpret this as evidence for in situ formation on largely
undisturbed orbits within massive, extended disks. Brown dwarf companions
exhibit a broad peak at e $\approx$ 0.6-0.9 with evidence for a dependence on
orbital period. This closely resembles the orbital properties and
period-eccentricity trends of wide (1-200 AU) stellar binaries, suggesting that
brown dwarfs in this separation range predominantly form in a similar fashion.
We also report evidence that the "eccentricity dichotomy" observed at small
separations extends to planets on wide orbits: the mean eccentricity for the
multi-planet system HR 8799 is lower than for systems with single planets.
(Abridged)
