Once upon a time, MSN messenger updated its conversation windows to allow emoticons in nicknames. A brief meme started where people titled themselves The [insert emoticon of choice]. I nicked The Dog from ahsirakh, who took up The Cat for a while. Then he got bored, as everyone else eventually did, and junked it.
Fads come and go. Somehow, however, The Dog hung around for a while. Then it stuck.