Tough question! I'd just say that "пёс" for a male dog is not as common as "кот" for a female cat.
Actually, as a native speaker, I'd hardly say "Тим - моя любимая собака" or "Тим - мой любимый пёс". I'd rather say "Тим - мой любимец" (Tim is my favorite/loved one), "Больше всего я люблю Тима" (I love Tim most), etc. With the cat named Tom, I don't feel inclined to use other phrasing. "Мой кот" sounds 100% ok.