Characters do not make the genus, but the genus gives the characters?
It has been a long time since, I've read The Origin of Species, but it sounds like the intent of the phrase is to suggest that the presence of similar characteristics in two species do not imply that they are related, but that related species would be expected to share characteristics. I.e. that similar characteristics do not imply relatedness, but relatedness implies similar characteristics. Genus -> Characteristics, rather than Characteristics -> Genus. These two possibilities are logically distinct.

The assumption that Genus -> Characteristics segues into the key concept of his theory of evolution, that related species (a genus) arise from the descent with modification of an shared ancestral population. Remember that Darwin wrote long before we understand the role of DNA as the carrier of genetic information. He knew nothing about the physical substance or the mechanism of transmission that passed characteristics to offspring, but through his observations built a strong case that such a substance exists and that gradual changes in this substance occur over time in a population. He later posits that natural selection leads to the accumulation subsets of descendants with modifications that are advantageous in their environment. Thus the species within a genus could arise, their similarities the result of their shared ancestry, their differences that result of descent with modification and selection.