I had the same question. According to Katzner E-R dictionary, животное (noun, animal) declines as an adjective. Животный is an adjective meaning animal or bestial. So I think the noun is the neuter form of the adjective.
I haven't seen why this adjective-noun connection is made and don't know if it's common.
Don't look for help from the notes - they're almost completely useless.
Here's a link to a noun-declension table I put together, which includes a chart of the Russian Spelling Rules:
I'm still working on a table for adjective endings
Also, see the collection my list of prepositions, their meaning(s), and the case(s) which their objects take:
No, they are just different words with different endings. In the nominative the former is "домашнее" and the latter is "животное". The letter before the "н" has no effect on the ending. For example it can be "страшное" ("scary") or "летнее" ("summer" as an adjective). It doesn't depend on any rule, it's just how they are.
They are not just different words - there is a reason for the difference. See my note below about hard-stemmed and soft-stemmed adjective endings (the stem "домаш-" ends in soft ш and the stem "живот-" ends in hard т). The н appears to be added to certain adjective endings, but I don't know why - yet.
Hard-stemmed vs. Soft-stemmed Adjectives
(For what it’s worth, hard-stem adjective end in hard consonants, while soft-stem adjective end in soft consonants. I'm developing tables for these endings, with some examples, but haven't finished that yet.)
I noticed that домашних and животных have different endings (-их vs. -ых) but they are the same case and number (neuter plural - gender is irrelevant to plural adjectives), and домашних is an adjective which modifies животных.
Why don’t they have the same ending? Why the different endings? In an article on adjective endings, I found some tables containing these plural endings:
In comparing these endings to домашних and животных, it seems that each has added "н", to make the endings -них and -ных. If that's the case, that helps explain the hard/soft distinction, because the stem for домашних would then be домаш- (ш is a soft consonant) while the stem for животных would be живот- (т is a hard consonant). It's just that none of the tables I've seen so far list the adjective endings as beginning with "-н".
Anyway, the hard/soft distinction is apparently why they have different endings - I'm pretty sure.
Also: the Russian Spelling Rules apply to adjective endings, so, e.g., -ым might be respelled as -им.
If your brain is exploding as is mine, and you want to be able to talk about animals and pets and nouns which are adjectives, singular different from plural rules turning accusative into genitive sometimes when only God and native speakers can know... just remember a few sentences and move on with it!!!
-ых or -их is the usual genitive plural and prepositional plural ending for adjectives. And remember that all plural animate words have the same form for accusative and genitive.
Here we have one adjective and one (animate) noun that started as an adjective (such that it still takes adjectival forms) so both words take -их in their accusative plural endings used here.
It's less about animate vs inanimate, and more about liking a specific object vs things in general. "I like pets" is a general statement, so it's "любить" (though "нравиться" is also correct, but, arguably, less common). "I like this dog" would be about a certain object, so it would be "нравиться" and using "любить" would mean "I love this dog".
This would be the same for people as well. "Я люблю умных людей" is closer to "I like smart people" because I am talking about smart people in general, but "Я люблю тебя" is "I love you" whereas "I like you" is "ты мне нравишься".
Essentially Russian doesn't make a distinction between "любить" and "нравиться" for general statements, but does for a specific object.