Analyzing this sentence linguistically requires us to ignore the particular name "Bingo" and the particular nouns "man" and "dog", and instead assume that these are generic entities that have the possession relationship, can be named, and are of the correct syntactic gender.
So let us convert this sentence into
There was a Klingon [who] had a brother and Krzakh was his name.
Now, is Krzakh the Klingon or the brother?
I would guess that it is the Klingon. The evidence is not strong, but the subject of the sentence is clearly the Klingon, not his brother, so the new information seems to point at the former. If I wanted to point explicitly at the brother, I could have used
There was a Klingon who had a brother, and his brother's name was Krzach.
I could also use either "and the brother's name" or "and the man's name". These two are parallel structures. But here's the thing; there is no parallel to "and his brother's name" except, well, "and his name". So my intuitive ear concludes again that, if "his name" is used, it is probably the Klingon.
With the actual nouns and names this logic may not work so well anymore, and the sentence can be interpreted both ways, which is unfortunate. If you don't write poetry (or try to amuse your readers in other ways), choose your words carefully to avoid ambiguity.