Recently, the possible co-existence of a first family composed of "normal" neutron stars with a second family of strange quark stars has been proposed as a solution of problems related to the maximum mass and to the minimal radius of these compact stellar objects. In this paper we study the mass distribution of compact objects formed in binary systems and the relative fractions of quark and neutron stars in different subpopulations. We incorporate the strange quark star formation model provided by the two-families scenario and we perform a large-scale population synthesis study in order to obtain the population characteristics. According to our results, the main channel for strange quark star formation in binary systems is accretion from a secondary companion on a neutron star. Therefore, a rather large number of strange quark stars form by accretion in low-mass X-ray binaries and this opens the possibility of having explosive GRB-like phenomena not related to supernovae and not due to the merger of two neutron stars. The number of double strange quark star's systems is rather small with only a tiny fraction which merge within a Hubble time. This drastically limits the flux of strangelets produced by the merger, which turns out to be compatible with all limits stemming from Earth and Lunar experiments. Moreover, this value of the flux rules out at least one relevant channel for the transformation of all neutron stars into strange quark stars by strangelets' absorption.