  We study the possibility of generating dark matter (DM) and baryon asymmetry
of the universe (BAU) simultaneously in an asymmetric DM framework which also
alleviates the small-scale structure issues of cold DM (CDM). While thermal
relic of such self-interacting DM (SIDM) remains under-abundant due to
efficient annihilation into light mediators, a non-zero asymmetry in dark
sector can lead to survival of the required DM in the universe. The existence
of a light mediator leads to the required self-interactions of DM at small
scales while keeping DM properties similar to CDM at large scales. It also
ensures that the symmetric DM component annihilates away, leaving the
asymmetric part, in the spirit of cogenesis. The particle physics
implementation is done in canonical seesaw models of light neutrino mass,
connecting it to the origin of DM and BAU. In particular, we consider type-I
and type-III seesaw origin of neutrino mass for simplicity and minimality of
the field content. We show that the desired self-interactions and relic of DM
together with BAU while satisfying relevant constraints lead to strict limits
on DM mass $\mathcal{O}(\rm GeV) \lesssim M_{\rm DM} \lesssim460 $ GeV. In
spite of being high scale seesaw, the models remain verifiable at different
experiments including direct, indirect DM search as well as colliders.
