  Multi-band observations of the fast radio burst (FRB) 131104 show that this
burst may be associated with a gamma-ray transient entitled Swift J0644.5-5111.
Follow-up observations for potential X-ray and radio counterparts of FRB
131104/Swift J0644.5-5111 got null results and provided the upper limits of the
emission flux at 5.5 GHz, 7.5 GHz, $U$-band, and X-ray band. By assuming this
association and using these upper limits, environmental properties (the
fraction of energyin a magnetic field $\varepsilon_{\rm B}$ and the number
density $n$) of the progenitor system of FRB 131104/Swift J0644.5-5111 were
constrained in the context of the standard afterglow model that neglects the
non-relativistic effect and jet effect by several groups. In this paper, we
adopt a unified afterglow model that takes into account the non-relativistic
effect and jet effect and use the upper limits of four bands (5.5 GHz, 7.5 GHz,
$U$-band, and X-ray) to obtain more stringent constraints on the parameter
space spanned by $\varepsilon_{\rm B}$ and $n$. We thus suggest that FRB
131104/Swift J0644.5-5111 might originate from a black hole-neutron star merger
event. Moreover, we calculate multi-band emissions from a kilonova powered by
the radioactivity of $r$-process elements synthesized in the ejected
neutron-rich material and find that the $U$-band emission from the putative
kilonova is significantly lower than the upper limit of the observations.
