This troper always thought they had one anus right in the middle of their body, it just wasn't visible (because, dude, it's a kids' cartoon). That's the theory that this troper sticks to because it's less squicky to her than the alternative (see above).
There was one episode which had the line, quote, "it has to pass through our colon," which verifies this theory.
This troper remembers an episode where they were snorkeling, and Dog makes a large fart bubble, which appears right in the center beneath them, verifying that this is where their anus is.
Possibly it would look like their bellybutton? It is in the right place to be either.
Well, at least in one episode they were shown peeing (although the camera zoomed away from the actual process, showing only their faces). It seems to be a mystery intended by the creators, since Dog spoofs it in another episode (I think it was an episode about a live-documentation about their species).
In the comics, Donald did have to repair his toilet bowl so...
We see Dog about to pee on a fire hydrant a few times, and he lifts his leg accordingly. So apparently they've got some sort of...exit on their body.
Lola figured that one out by barging into their bathroom. They have a LitterboxFireHydrant.
There's this one episode about...I don't know, something about eating habits, and a goldfish, and long story short, Cat disappears into Dog's mouth. I saw it when I was a kid, and I have literally not been able to figure it out for YEARS, because, get this, at some point, Cat comes OUT OF HIS OWN MOUTH, INSIDE OF DOG'S STOMACH. Just...how?!
Here's another one, how is it possible for Rancid Rabbit to have all these jobs and scam Cat so easily and he's the smart one. I was hoping someone would see Rancid as a restaurant and ask Catdog, "Hey Catdog, why is your doctor our waiter?" The only time Cat notices this is when Rancid appeared as a teacher and a Policeman. Also in "All you can't eat" why was Rancid crying when he got fired as Manager from Taco Depot when he has a hundred more jobs.
How is it that nobody at all remembers that CatDog won in a lot of the episodes? In literally half of the episodes, they win or are otherwise unharmed, so how is it nobody remembers those episodes?!