  The localization of the repeating FRB 121102 to a low-metallicity dwarf
galaxy at $z=0.193$, and its association with a quiescent radio source,
suggests the possibility that FRBs originate from magnetars, formed by the
unusual supernovae in such galaxies. We investigate this via a comparison of
magnetar birth rates, the FRB volumetric rate, and host galaxy demographics. We
calculate average volumetric rates of possible millisecond magnetar production
channels such as superluminous supernovae (SLSNe), long and short gamma-ray
bursts (GRBs), and general magnetar production via core-collapse supernovae.
For each channel we also explore the expected host galaxy demographics using
their known properties. We determine for the first time the number density of
FRB emitters (the product of their volumetric birthrate and lifetime), $R_{\rm
FRB}\tau\approx 10^4$Gpc$^{-3}$, assuming that FRBs are predominantly emitted
from repetitive sources similar to FRB 121102 and adopting a beaming factor of
0.1. By comparing rates we find that production via rare channels (SLSNe, GRBs)
implies a typical FRB lifetime of $\approx$30-300 yr, in good agreement with
other lines of argument. The total energy emitted over this time is consistent
with the available energy stored in the magnetic field. On the other hand, any
relation to magnetars produced via normal core-collapse supernovae leads to a
very short lifetime of $\approx$0.5yr, in conflict with both theory and
observation. We demonstrate that due to the diverse host galaxy distributions
of the different progenitor channels, many possible sources of FRB birth can be
ruled out with $\lesssim 10$ host galaxy identifications. Conversely, targeted
searches of galaxies that have previously hosted decades-old SLSNe and GRBs may
be a fruitful strategy for discovering new FRBs and related quiescent radio
sources, and determining the nature of their progenitors.
