What exact characters determine an animal's Genus?
I don't think there is any non-arbitrary concept being used to tell whether two species belong to the same genus or not. The same goes on with any other taxonomic rank. Systematicists build some intuition within each clade about what a genus could mean and use it to argue what is the most strategic grouping but the concept of genus may well vary a lot from clade to clade (from spiders, to mammals or to flowering plants, ...).

Note that even the concept of species is not as nicely defined as you may think. Have a look at How could humans have interbred with Neanderthals if we're a different species? for discussion on the concept of species.