i'm not sure if unconventional is the right term but anyway

i'm looking for japanese dramas & movies that highlights certain ideas, principles, or circumstances in our society that are often overlooked, unknown, underestimated, or unaccepted.

some examples would be:
koisenu futari: aromantic asexual people
kahogo no kahoko: sheltered children
shoplifters: family by choice, not blood.

koisenu futari really got me real good these days, and i'd love to watch more stuff that represents the minority! :<

preferably not centered on BL or su*c*de since they're already a common topic.
also no cohabitation, unless the people involved are unconventional (like in koisenu futari).

"Found family" (like the one in Shoplifters) is actually a pretty common trope in Japanese dramas. If you don't mind watching other dramas with that trope, I highly recommend Quartet and anone.

Some other stuff that I think quite "unique", whether in their subject matter or their approach:

  • Watashi no Uchi ni wa, Nanimo Nai (follower of minimalist lifestyle)
  • Kataomoi (spoiler: transgender identity and how hard it is to get it recognized "officially")
  • Saka no Tochu no Ie (postnatal depression)
  • Kawa no Hotori de (homeless people, but it's told as a slice-of-life drama; the characters are very chill; it's not sad and it doesn't try to "educate" the viewers about the plight of homeless people)
  • Kakenai!? (about writer's block; a pretty common condition but I can't remember it being the central focus in any drama/movie/book ;p)
  • Hoshi to Lemon no Heya (about hikkikomori/shut-in--this is a drama special/TV movie)

Not sure it will fit your criteria, but here are some other shows I could think of (not always treated in a realistic way though): 

- Freeter, Ie wo kau  : how to buy a house when you're working part-time
and on the same topic : Princess Maison

- adoption (from a teenage mother): the movie True Mothers

- raising an impaired child (it's somehow a cohabitation show but it's more than that) : Kiseki no Hito

- taking care of the deceased and their family : the wonderful movie Departures or the family business drama (but also mystery genre) Saikou no Jinsei no Owarikata

Japan is also pretty good at making dramas anticipating social issues consequences (and how to solve it - dystopian dramas): 

- declining marriage rate: Kekkon Aite wa Chusen de

- environmental issues: Nihon Chinbotsu: Kibo no Hito

And of course you've got countless of dramas centered on illnesses,  or on jobs (and not always the most known ones like cook in a school, supermarket employees, healthcare employees...)

Yuube no Curry, Ashita no Pan (2014)- in-laws living together as a family even though the husband/son has already passed away 7 years ago.  It's a story of moving on and also has plenty of cooking/eating scenes.

Utsukushii Kare - quite different than other classic rom com japanese BL. Have some dark undertone

37 Seconds - journey/story of a disable manga artist

Kinou Nani Tabeta? - closeted late 30 year old gays living together ( a very realistic portrayal of LGBTQ in japan)

And my absolute favorite To Each His Own - finding hope in unconventional way

Iki mo Dekinai Natsu- It is about a woman who learns that she was not on her family register. Also talks about the effects of not being on one (job, citizenship, etc.) 

26 Years Diary- It is about a Korean man studying abroad. 

3 Nen A Gumi: Ima Kara Mina-san wa, Hitojichi desu- The effects of social media (high school)

Aishiteriru to Ittekure 1995- Deaf male lead (physical disability)

ATARU- Male lead is on the Autistic Spectrum 

14-sai no Haha- Teen pregnancy 

Bittersweet- Female lead learning her friends sexual orientation

Bloody Monday series - Terrorist Attacks being the main focus

Close-Knit- Transgender lead, found family and neglectful parent

Dandarin Rodo Kijun Kantokukan- Labor Standard Inspectors as main focus

Final Fantasy XIV: Hikari no Otousan- father and son reconnecting through video/computer game

Who Are You  (bullying at school)
Angry Mom  (bullying and school + teenage suicide)
Mother (child abuse)
I Love My President Though He's A Psycho  (houshold abuse)
Hajimemashite, Aishiteimasu  (child abuse -+ neglecting)
Hokuto  (child abuse)
Iguana no Musume  (verbal abuse in family)
Woman  (single mother in need)