I mean, for the purpose of playing the devil's advocate, they have to refer to themselves as something if they have a concept of themselves, right? Even it's something as simple as a chimpanzee thinking "I'm hungry," or the chimpanzee's equivalent thought that he has developed as a pondering of the grumbling feeling in his stomach, it has to call itself something. Hell, if dogs can respond to being called by their name, a chimp can sure as hell create a name for itself.
No, not necessarily. They have no concept of language. Language is such a human thing, it's integrated into our very thoughts. Animals have no such connection to it. There's no reason for them to assign objects names, or their actions verbs. They only need to assign vague concepts, like whether something is edible, or what a tool can be used for.
And, as far as I know, dogs don't recognize themselves as being named whatever the owner names them. They just respond to the sound, like they respond to any other command they've learned to recognize.