Some interacting supernovae (SNe) of type IIn show a sizeable continuum polarisation suggestive of a large scale asymmetry in the circumstellar medium (CSM) and/or the SN ejecta. Here, we extend the recent work of Dessart et al. on super-luminous SNe IIn and perform axially-symmetric (i.e., 2D) multi-group radiation hydrodynamics simulations to explore the impact of an imposed large scale density asymmetry. When the CSM is asymmetric, the latitudinal variation of the radial optical depth $\tau$ introduces a strong flux redistribution from the higher-density CSM regions, where the shock luminosity is larger, towards the lower-density CSM regions where photons escape more freely --- this redistribution ceases when $\tau<$1. Along directions where the CSM density is larger, the shock deceleration is stronger and its progression slower, producing a non-spherical cold-dense shell (CDS). For an oblate CSM density distribution, the photosphere (CDS) has an oblate (prolate) morphology when $\tau>$1. When the CSM is symmetric and the ejecta asymmetric, the flux redistribution within the CSM now tends to damp the latitudinal variation of the luminosity at the shock. It then requires a larger ejecta asymmetry to produce a sizeable latitudinal variation in the emergent flux. When the interaction is between a SN ejecta and a relic disk, the luminosity boost at early times scales with the disk opening angle -- forming a super-luminous SN IIn this way requires an unrealistically thick disk. In contrast, interaction with a disk of modest thickness/mass can yield a power that rivals radioactive decay of a standard SN II at nebular times.