The "other cat" is D.B. Cooper (short for "Ding Bat" -- Seriously!) She belongs to Holly's father, Pete, and has basically hated Max since first sight. Actually, from their first contact, Max learned how to cuss. Oddly, he always just wanted to be friends. Didn't work out.

From the hallway at Pete's place. DB is under the bed, hoping Max will go away.

The dog is Polly, Dave's parents' dog (formerly Dave's dog). She's about 14 or 15 years old, and doesn't (excuse me) give a rat's ass about Max. She will basically ignore him all the time. On the other hand, Max will pretend to ignore Polly, but will, of course follow her around to make sure he always knows where Polly is. Here he is at Christmas pretending to ignore Polly. Note how Polly is genuinely ignoring Max.