Both Main Characters are pioneers, e.g. in the banking industry, and support the education of women. Both dramas are depicting the life story of a real person.

Both dramas deal with the same time period,."Yae no Sakura" is somewhat a precursor to "Tsuda Umeko...", the drama's plot starts earlier in history and catches up to the period in the movie and goes beyond. Both female leads are working for the cause of women's education and are influenced by western culture. One of the companions that studied abroad with Tsuda Umeko is a side character in the drama: Yamakawa Sakiko / Oyama Sutematsu has a pretty interesting story arc, the viewer feels compassionate with her alienation towards her homeland and the reaction of her Japanese countrymen, her engagement/the relationship with Oyama Iwao gets shown in more detail and thus seems reasonable and pleasing.

In both dramas college girls sell themselves/fake a relationship etc.
"Ashita, Watashi wa Dareka no Kanojo" just started airing, so currently the SFL is revealed to do papakatsu (including more than companionship - as seems to be the norm, according to the other character's reactions in the drama and documentaries on youtube, contrary to what the "Papakatsu" synopsis implies), whether or if the FL does it too will be revealed in the upcoming episodes... so far the drama seems to show a more in-depth portrayal of the psychological effects and dynamics surrounding that business/lifestyle as you see it from many character's perspectives. While the FL in "Papakatsu" seemed more naive and romantically inclined and the story revolved around the two main characters, the FL in "Ashita, Watashi wa Dareka no Kanojo" is totally aware of the whole business situation, is on top of things, and has multiple clients.

both have the same subject. I imagine "Akiko's piano" as a detailed account of what happened with one of the pianos in "Mother’s Bomb-struck Piano", kind of a prequel.

Yae is born into the Aizu clan. Being the daughter of a samurai family she comes in contact with guns during her childhood and passionately hones her gunnery skills against everyone’s opposition (not fit for a woman). Later she actively participates in the Boshin War. The political events during that time are depicted in depth.