Context. The latest Gaia Focused Product Release (FPR) has provided variability information for ∼1000 long-period red giant binaries, including almost ∼700 ellipsoidal binary candidates, thus providing the largest sample to date of this binary type with both photometric and spectroscopic time series observations. Aims. We aim to characterize both physically (luminosity, mass, and radius) and chemo-dynamically (metallicity, [α/Fe], and Galactic velocities) the population of long-period red giant ellipsoidal binary candidates and a subsample of rotational variable candidates from Gaia FPR by combining Gaia astrometry, photometry, and spectroscopy observations. Methods. We crossmatched the Gaia DR3 measurements (positions, velocities, atmospheric parameters, and chemical abundances) with the catalog of long-period red giant candidates from the Gaia FPR, which has photometric and radial velocity variability information. Combined with the photo-geometric distances, we estimated the extinction, bolometric magnitude, luminosity, spectroscopic radius, and mass. The accuracy of this method was tested for similar samples in the literature, including red giant binaries that have asteroseismic-determined physical parameters. Results. Ellipsoidal variables are characterized as being low- to intermediate-mass stars (0.6 ≤ ℳ1 ≤ 5.0 M⊙) with radii as large as the Roche lobe radius of the binary. Eccentricities tend to be lower for primary stars with smaller radii, which is the expected result of tidal circularization. Combined with the orbital properties, estimates for the minimum mass of the companion agree with the scenario of a low-mass compact object as the secondary star. There are at least 13 ellipsoidal binaries with orbital periods and masses of the two stars compatible with model predictions for Type Ia SN progenitors. For the rotational variables, their orbital periods, enhanced chromospheric activity, smaller radii, and low mass (ℳ1 ≲ 1.5 M⊙) point to a different type of binary than the original ellipsoidal sample. The Galactic velocities indicate that ellipsoidal variables are found both in the Galactic disk and halo, while rotational variables are predominantly concentrated in the Galactic disk. The velocity dispersion is much higher in the ellipsoidal than in rotational binaries, probably indicating older dynamical ages. The enhanced [α/Fe] abundances for some of the ellipsoidal binaries, having ℳ1 ≥ 1.0 M⊙, resemble the population of young α-rich binaries in the thick disk. An episode of mass transfer in those systems may have produced the enhanced α abundances and the enhanced [Ce/Fe] abundances reported in a few ellipsoidal binaries. Conclusions. Luminosities, radii, and masses were derived for 243 ellipsoidal and 39 rotational binary candidates, composing the largest Galactic sample of these variables with chemo-dynamical and physical parameterization. Based on their mean chemo-dynamical properties and stellar parameters, these binaries can be considered as two manifestations of the same phenomena, a close binary with a giant primary, instead of two independent and unrelated binary types. Detailed future analysis of individual sources will provide insights into the history and future evolution of these binaries.