Yes, and if we just accept arguendo that such black holes exist in our universe (and in reality they have not been precluded, as I note below), we have a density problem, although the argument is pretty Copernican (or alternatively rests on the cosmological counting system: forbidden, exactly one, mandatory).
One problem with the argument that there is just one in the solar system is that unless it's unique, we would expect to see effects on star motions on a large enough number of nearby stars that one has to be inventive in attempting to explain why we haven't notice that yet. Worse, you would expect that at least one PBH would be near each supernova, and would interact with remnants, and that does not appear to be the case. Additionally, even if there was one in essentially every star system in every galaxy, they're still a tiny fraction of the dark matter sector, and we'd have to work even harder on an explanation for the core-cusp problem.
That said, ~ the high end of sublunar mass PBHs (~ 10^23 kg) is about the least constrained mass region, so it's not surprising that people speculate about ~ M_earth PBHs. "[H]ow it came to be associated with our solar system" is fairly straightforward -- the field of PBHs existed before large structure formation, so you have the question a little backwards. The PBH was here first and its perturbations of the pre-sun GMC would have influenced the formation of our solar system. The question is whether those perturbations conflict with reasonable hypotheses for the formation and evolution of the solar system (and its predecessor). And again, you would expect PBHs to be involved in most similar star systems, and get to wonder about the gradient of such tiny black holes from the centres of galaxies to space outside galactic clusters. (Extragalactic space inside galactic clusters is maybe worse: there's invisible mass-energy among the galaxies mixed in with dust and gas, so why aren't stars forming there if the density of small PBHs is roughly similar to the outer edges of galaxies within the clusters? And if the density is very different, then why?)
Finally, https://arxiv.org/abs/1607.06077v3 is very fresh and lists some further relevant constraints, but leans especially on https://arxiv.org/abs/1301.4984 .