So far, it doesn't look like MCCC is selecting our "undeniably unique pooches and padfoots," as @VeeDub so eloquently described them.
First, we loaded up one of our existing saves. We sent a Sim to a Cat Café, figuring a feline might be more likely to spawn here than anywhere else. A kitty turned up fairly quickly, but it was a plain ol' purrer called, simply enough, Cat. We'd seen her before.
We figured the game might have already pre-selected all the strays it would ever need, so we decided ty try again, with a brand-new save. Our protagonist went for a walk around his Newcrest habitation, and spotted two stray cats.
One was Cat, whom we've met a few times before. Another was a pretty white kitty named Doodle. Yellow Kitty, Fancy Kitty, and Schmancy Kitty were nowhere to be seen.
Next, we sent our Sim upstairs to his computer, where he browsed the listings of adoptable cats and dogs. There, we saw a full cast of adorable pets looking for homes...but none of our creations were among the possible choices.
We haven't been searching long, so it's very possible we could still see one of our Gallery friends. We also haven't seen a stray dog yet. But, for now, it looks like we have a better chance of seeing Abraham Lincoln working as a yoga instructor or a paparazzi.