This is an educated, albeit only a speculation of mine that I never confirmed: in the past (medieval times) people used to describe inferior things by comparing them to dogs like "dog's weather", "you will die like a dog" and many other sayings. It's wrong to think that your mother is being insulted by being called the "son of a bitch" (it's also pointless to censor the word as you did). What it means, or at least what it originally meant, is that you came out of a womb of a dog.

In other words, if you're with your mom in a room, and someone calls you a son of a bitch, the claim is not that the woman you take for your mom is a bitch, rather, that you're wrong and your mom is actually not in the room (unless there's also a female dog in that room).