I'm looking for Japanese or Korean dramas featuring unlikely friendship. You know, where they have these people who are very different in age, social status, outlook in life, etc, but end up being friends anyway. Something along the lines of "My Mister" or "G Senjou no Anata to Watashi" (with Yukie-san and her two younger friends).

My only requirement is that this friendship should remain platonic. I'm okay with romance as long as it's merely hinted at and not made explicit.

Any recommendation is appreciated :)

I'm unsure if you are looking for just friendship between two characters. Or friendship between the male lead and the female lead (as long as it stays platonic). So I'll give you a mixed list:

  • Spring Turns to Spring | Friendship between FL and 2FL. One is a greedy reporter with a broken moral compass. The other is a former actress who cares about family and righteousness, but who's been cheated on by her husband. These two characters end up switching bodies and chaos ensures!
  • The Guest | The three main leads get tied together by a common tragedy. One becomes a priest, the other a shaman and the third a cop.
  • The Last Empress | FL and 2FL start out as enemies but end up becoming friends towards the end, both trying to bring down the same family.
  • Terius Behind Me | ML (former secret agent) becomes the FL's children's nanny! Though he takes the job for an ulterior motive, he builds a great friendship with the FL and her family.
  • Queen: Love And War | Friendship between the FL and a support male character. You find out they come from very different backgrounds, but they are close friends.
  • Woman of 9.9 Billion | Friendship between two women, one poor and the other rich. The friendship is surrounded by mystery. The FL also develops a friendship with the ML with some hints of romance.
  • Age of Youth | The four main leads are all very different but they come together in a beautiful friendship. First episode is painful, but it gets better after that.
  • Cheese in the Trap | Friendship between the FL and 2ML is interesting.
  • Hotel del Luna | Although it ends in romance, the friendship between the ML and 2ML is well constructed. Both characters are opposite to one another and learn different things from each other.
  • Kill It | ML is a hitman, FL is the cop trying to take him down.
  • Oh My Ghostess | Friendship between the FL and 2FL. One is a ghost, the other a sort of medium. They start on the wrong foot but build an understanding and then a friendship from there.

i guess something more along the line of Slice of life right

- kekkon aite wa chusen de -  (set in a fictional setting where government set you on a blind date, the ml is an unpopular guy whos desperately trying to get married, the fl is a beauty and popular woman whos for some reason trying to avoid marriage) - the romance part doesnt take the majority of the story tho, its more about life.

- sagideka - a police and a fraud, no explicit romance but its clear theres strong bond between them. the police has somewhat more sympathy toward criminals than average people.

- the confidence man jp - about 3 conman (and conwoman ?) - a witty woman, an emotional guy, and an oldman. its an episodic comedy, but believe me, every episode always surprise me.

Ashita no Kita Yoshio

Sagideka +1

Films:

A Long Walk

Honokaa Boy

  • Just Between Lovers - the friendship / found-family relationship between the ML and his loan shark, who essentially becomes his adoptive grandmother.  
  • Mr. Sunshine - the unlikely friendship / camaraderie between the ML, SML and TML. They all come from very different backgrounds and are basically rivals, but sometimes their interests align and at times they drink or silently commiserate together because they understand each other. 
  • Mad Dog - the relationship between Kim Min Joon and Choi Kang Woo. They start out as basically enemies / rivals but eventually their relationship becomes unexpectedly close like a father and son or older and younger brother.

Chief Kim - This is more of a typical bromance between two guys with very different personalities, but there are so many beautiful friendships in it apart from the main leads'!

Smoking - Here we have a found family group with father/son, mentor/mentee, and various other friendship tropes.

The Guardians/Lookout - People growing closer because of shared tragedies, including Enemies To Friends trope, Big Sister & Little Sister, Big Sister & Little Brother, and many more.

Bunny/Usagi Drop - The relationship between the main leads here probably doesn't really count as "friendship" since they are parent & child in every way that counts, but it's still a beautiful relationship to watch develop!

I wish I could think of more recommendations from the top of my head.

The Untamed : Between 2 ML. There are many different interpretations of their friendship but it perfectly fits your description.

Goblin  - My all-time favorite bromance. Goblin and the Grim reaper relationship is way better than the actual romance in this drama. 

A Korean Odyssey - You will love the 'monkey' and 'cow' moments. The only reason I rewatched it 3 times