The meaning of family

all kinds of beautiful family-like relationships and friendships between non-blood related people; including (but not limited to): a friendship between a woman and her abused young neighbour, where they behave just like mother and son; brother-like relationship  between a boy and his driver; friendship for life from multiple teenagers; subplots like patchwork families etc.

a man takes care of his dead best friend's daughter and mother

"Family is forever" - Mr. Wang
Just watch it, can't say anything else without spoiling

three women become best friends and help each other (plus a stepson to one of them) feel the familial warmth they long for

depressed man picks up a woman and a child and forms a temporary family with them

found family

enemies to family-like relationship where Zhang Yi takes care of an orphaned teenage girl

enemies to family-like relationship where Zhang Yi takes care of an orphaned teenage girl

found family 

a soldier picks up an orphaned child and raises him as his own

woman takes care of her best friend's child and enters a contract marriage with his brother to protect the child

FL takes over someone else's body and has to take care of "the other" woman's family

three sisters take in their youngest half-sister

a man assumes identity of a dead man and takes care of his wife and child

4 people pretend to be family for the sake of their espionage operation

adopted daughter

FL takes over someone else's body and has to take care of "the other" woman's family

two guys (one older, the other younger) bond over money and form father-son-like relationship

mother and son befriend an elderly lady and form a family-like relationship with her

a woman kidnaps a young girl to protect her from her abusive parents; loose remake of Mother

Kim Hye Soo runs a crime organisation with her adopted "daughter" Il Young

stepmother

a man takes care of son and mother of his deceased boyfriend

amnesiac man becomes a "fake" brother

a female bodyguard is tasked with protecting and raising a young prince

three fugitives take care of a little girl

a young boy picks up an orphaned toddler (warning: sad film)

4 women, 2 men and 2 infants come together to form a co-family

found family; unrelated people living together (almost) like a family

father-children like relationship

mother-son like relationship

father-son like relationship

found family

A group of strangers forms a foster family to help little girl regain her memory 

A politician takes in his ex-crush's kids

ML takes care of his "other timeline" self's kid and wife

ML gets reincarnated into a body of someone else and has to take care of "the other" family (wife and kids)

an assassin and a lonely young boy become father and son

father searching for his long lost son and a young man searching for his parents form a father-son like relationship

ML has to take care of wife and kids of a man whose identity he assumed (and more)

father/son-like dynamics embark on a roadtrip looking for boy's mother

woman finds a baby on the street and takes care of it

disclaimer: it's rubbish and I do NOT recommend it but I put it here because it fits the theme. Watch at your own risk. 

A group of people move in together and become a multi-generational big family

Sequel to The Aobas' Dining Table (webdrama), focuses more on friendship and coming of age rather than found family but the living arrangements from the previous part are still relevant

A widow takes in an abandoned kid

an orphan girl becomes sworn sisters with a princess and gets adopted by the emperor

Set during 1960's when children from poor families were sent to Mongolia to be raised; adopted children/family not related by blood

former delinquent takes care of a distant relative's neglected child

four children live on their own

young actress takes care of an old farmer; they find a wounded man and begin living together

an old woman takes in a young man who saved her life

three estranged half-sisters 

subplot; one of the characters takes care of his dead friend's daughter

kind of

ML pretends to be someone he isn't; has to live with his fake wife and family

woman stumbles upon an amnesiac young man and decides to take him in; also features other relationships like stepmother-stepdaughter

family-like dynamics

romance between two men; focuses on daily life and two MLs co-parenting one partner's younger brother

father-son like dynamics

family with a pinch of Stockholm Syndrome

a family life of a man, his daughter, and his dead brother's husband

man picks up an orhpaned girl

Stockholm Syndrome family version

a group of people pretending to be family

a woman transmigrates and has to take care of her "other timeline" stepchildren

uncle takes in his niece

two friends (prepare to) co-parent a baby

a man "adopts" a child and her older (adult) sister; it focuses more on romance and is an adoption on paper only; not that good (I do not recommend it)

two boys got mixed up as babies and are raised by families that are not their own

an aunt takes care of her niece

I 2xed through this so I don't remember details but FL takes in two kids that are not hers