Every single noun in Danish has a grammatical gender. There are two grammatical genders in Danish: common gender (-n words) and neuter gender (-t words).
For example, katten (the cat) is a common gender noun because its definite form ends in -en. You would also say en kat (a cat).
Dyret (the animal) is a neuter gender noun because its definite form ends in -et. Et dyr (an animal), dyret (the animal).
In general, animals/living things will be common gender nouns (nouns which definite forms take an -en at the end, e.g. hesten, katten, fuglen), but sometimes they will be neuter gender (hamsteret, dyret). So it depends on whether or not the word is common gender, neuter gender, or even plural, not how they can be grouped together.
Neither of those reason are correct. „Katten“ means „the cat“, where simply „kat“ means „cat“. And also in Danish the „is“ is nit required in this case. Both „Katten spiser dets mad“ and „Katten spiser sin mad“ mean „the cat eats/is eating its food“ but the use of DETS indicates that the cat is eating another „it“s food; SIN however, indicates that the cat is eating „its /own/“ food. The latter (using DETS) would be a gender neutral version of „Katten spiser hans mad“. The reasoning is also why SIN/SIT/SINE are only changed based on the article used for the particular word and whether the word is plural. I hope this clarifies things~