Is that his girlfriend or his dog he is singing to?
*Remember when you ran away .... You thought it was a joke And so you laughed You laughed when I said That losing you would make me flip my lid ....a I cooked your food I cleaned your house And this is how you pay me back For all my kind unselfish, loving deeds Ha! Well you just wait They'll find you yet and when they do They'll put you in the A.S.P.C.A. You mangy mutt..." *
Maybe he was calling his girlfriend a bitch??
It is hard to tell because it is literally a song about complete insanity. He could be singing to his mother or absolutely no one at all other than his imagination. I certainly never guessed he was singing to his dog but I suppose that is as possible as anything else.
I always thought it was his dog. The beginning of the son it sounds like a girlfriend, and he’s having some anger issues with his girlfriend laughing at him, gas lighting him, and that he’s metaphorically losing his mind.
But its clear to me, by the end, that he’s always anthropomorphized his dog in a manner only an actually insane individual would.
Thread from earlier this year: http://boards.straightdope.com/sdmb/showthread.php?t=823152
Also, springing it on us, after most of the song was delivered, in such a casual matter-of-fact way, means that this particular delusion was there all along, and we’re all just so oblivious to the reality – but that’s our problem.
But Shagnasty:'s right, its a totally unreliable narrator, it could be any option, none, or something else.
Did the songwriter ever have a follow-up?
Ah, the good old days of music that was weirdly funny, and made it onto your local radio station. 
Haven’t heard that song in decades, but I can still quote the thing verbatim.
You can draw your own conclusions about what that says about me.
Sometime around 1980, DC-area DJ Cerphe did a Saturday night show of nothing but comedy and offbeat stuff (including stuff like “Stairway to Gilligan’s Island” and “A Day in the Life of Green Acres”) that started off with:
His dog. He’s not as insane if it’s his girlfriend.
That’s what I’ve always assumed, and what I thought was part of the whole joke with the ending line.
Both-His mangy mutt was his girlfriend.
You might enjoy this: https://m.youtube.com/watch?v=tKq5nv9G68c
Jerry Samuels is … still alive!
I assumed it was a woman…since I’ve got a cassette at home by Napoleon XIV with this song, and on one of the other songs on the same album is “I’m happy they took you away, ha-ha” by Josephine XV. It’s essentially the same song told from her point of view.
I don’t know if it’s a follow-up or not, but another song of his drives home the fact that he’s just bat-shit crazy:
There used to be a radio program that aired late at night that would play weird songs like the one in the OP. I forget the name of it… I believe it was hosted by a man who called himself Dr…(can’t remember)
Anyway, “They’re Coming to Take me Away” used to be one of mt faves.
Dr Demento
Thank you! That was driving me crazy! (Ha Ha…)
There was a follow up - I’m Happy They Took You Away by Josephine XV.
So definitely a girlfriend.
There was apparently another followup in 1998, fascinating. In this one he refers to the subject as his pet, mentions cleaning its cage, and threatens to swing it around by the tail.