The only one I can think of that I have watched that features an animal at all prominently is Because This is Our First Life. The cat is pretty adorable and definitely gets some attention. But wow this is a hard one for me! I'm looking forward to hearing all the other recs!

The three baby cats/puppies in Marriage Contract really did well with providing more avenues that gradually melted the male lead's cold persona.

Meow, the secret boy, The man’s voice both have fantasy elements where the cats have human qualities, the Let’s Eat series has dogs as prominent characters in each season. I only watched 2&3 so not sure of how dominant the dog was in season one. Level up has a turtle which is a special friend to the male lead.

For the cat lovers, try Meow: The Secret Boy (Sol-ah is a cat who turned into a boy/guy) and even Imaginary Cat (Bokgil). Other Kdramas for animal lovers: Kill Me, Heal Me (golden retriever dog), Romance Is a Bonus Book (another Kdrama featured golden retriever dog!), Kill It (featured so many dogs), He Is Psychometric/That Psychometric Guy (another Kdrama for dog lover featuring a lost puppy) and The Secret Life of My Secretary (arowana pet fish).

Ahn Hyo-seob's character had a pet chick in Thirty but Seventeen. And then there's Gae-hoon, the stray dog turned family member in Link: Eat, Love, Kill.

Uhhh… Maximus the horse from The King: Eternal Monarch? As far as I’m concerned, the entire show was about him. Idk any of the humans involved.

It's been a while, but my first K Drama, "Let's Eat" had the cutest small dog that played a meaningful part in the story!

The Sapsali dog in Chicago Typewriter - was adorable and kind of important to the story. The choice of dog breed was also deliberate which you can understand once you watch the show. In fact there's a Sapsali in Alchemy of Souls too and for similar reasons.

Not a Kdrama but a Jdrama, if you're interested: Juui Doolittle stars Oguri Shun as a Dr. Doolittle type vet. Every episodes is basically a PSA on behalf of animals.

Military Prosecutor Doberman
Meow, The Secret Boy
Variety Show: Dog's are incredible (They train dogs who are tough to raise)

Wok of Love began with a talking horse and then discarded him in order to become a mediocre drama instead of the wildly weird and interesting one it started as.

Those recommending Meow: The Secret Boy as a drama for animal lovers might want to think that one through...
That would be a whole 'nother post.

Unless you're recommending it as a drama about an experienced and skilled Cat actor who taught his human costar a thing or two. Where are you, Cat Sunbae? I hope you haven't gone to the Great Acting School in the Sky.

Kim Myung-min had to act with not one but two large Collies - in Beethoven Virus and A New Leaf. Apparently he actually was a bit scared of them, cos he'd had a bad experience with a fierce dog when he was a child.

It is funny how the first drama that came to my mind was Mama Fairy and the Woodcutter. The drama had some animals as far as I remember. It took me long enough to remember the name of the drama, because this was such a weird fantasy drama.

oh my ghost, but thats creepy
Coffe prince (dog)
Masters sun (ghost dog)
You are beautiful (Jolie)
Answer me 1997
Father is strange (but technically, chihuahua is... I dont even know what it is)
Scent of a woman
That psychometric guy
Imaginary cat
The smile has left your eyes (psycho and his cat)
um, if Gumiho counts, then, My girlfriend is a gumiho & My Roommate is a gumiho
Shopping King Louie has a centipede, well, not for very long, but...
Go back spouses (stray cat)
Who are you? (2008) had a turtle
Graceful family (fish, but dead, but hilarious funeral)

I know I see a lot of heroines looking after or trying to connect to stray cats but I can´t name that many

Do Do Sol Sol La La Sol
The Beauty Inside
FLs crooning over their dressed up fluffy dogs plus they do not stop appearing after an episode or two.

The Man’s Voice has a cat (that can talk to the heroine). It’s a web drama and I started it because I was like “well if it’s terrible it’s only short” but it was better than I expected it to be.

1. Do the mice/rats in Mouse count?
2. The sad dead ghost dog in From Now On, Showtime
3. As above, 2 in Tomorrow
4. And wasn't there the best little dog in Nine Tails that went from wounded and scruffy and turned into an entirely different breed once it was rescued (and that's not to count those with two legs and 9 tails). I'm guessing the Imoogi doesn't count either, but shouldn't we be inclusive?
5. My all time fav animals are the mythical sacred tiger cubs in the c drama Sword Snow Stride. They are mine. All mine. Snarky little things.

Some Japanese dramas on Viki that have pets are Only Just Married (a cat that usually is being held) , Why I Dress Up For Love (a dog that they interact with a lot) and Princess Jellyfish (which actually has jellyfish in an aquarium).

I think tomorrow is good example. It had a complete episode about a sweet dog who wanted to commit suicide and our heros saved the dog. Great one for an animal lover and everyone looking for something emotional 😭

Woori from This is my first life.
Barasshi from Let's Eat.

So I thought about one, and then decided not to mention it. And then decided to say ok, I'll mention it. Imaginary Cat. There you go. Those that have seen it can see my dilemma.

The tiny turtle in "Love Me if you Dare". He even carried some flowers.