This work concerns weakly confined particle systems in the plane, characterized by a large number of outliers away from a droplet where the bulk of the particles accumulate in the many-particle limit. We are interested in the asymptotic behavior of outliers for two classes of point processes: Coulomb gases at determinantal inverse temperature confined by a regular background, and a class of random polynomials. We observe that the limiting outlier process only depends on the shape of the uncharged region containing them, and the global net excess charge. In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. For a finitely connected uncharged region $\Omega$, a family of limiting outlier processes arises, indexed by the (Pontryagin) dual of the fundamental group of $\Omega$. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points. The latter result is a manifestation of screening properties of the particle system.