Just to add to the conversation of answers here,

I think in very contemporary context, what a lot of modern Television takes from the original and often day-time AMERICAN TELEVISION soap operas, is not just this idea of interpersonal relationships, but also characters and the reactions to plots that tend to be VERY "melodramatic".

The following link is a article of a paper I found on the posisble correlation between Soap Operas and the Melodramatic, as the earliest forms coming from British Radio Soaps were often based in [British social] realism and make no mention of melodrama, when that argument is then counteracted when referring to American Prime Time Television with the likes of Dallas. In other words, I think the definition changed over time and through cultural (British to American) and medium (Radio to Television) differences and we now have both the original Soap Opera Realism & The Soap Opera Melodrama.

And as others have mentioned, it's these melodramatic elements are what allow many RECENT shows (Supergirl, Lucifer, Gilmore Girls) that are often sited to be primarily set in other drama genres (fantasy/superhero, crime/family, family) to still have "soapy" elements (interpersonal relationships and heavy romance) within the series and they often then also have a hybrid serialized and episodic structuring, which may also then go back to the original definition.

Some more recant contemporary examples of mixed genre shows with "Soapy" elements are

The Good Wife - Often toted as an "American legal & political Drama", is really a legal, political, family Dramedy & melodramatic romance. It's also a serialized-hybrid having both ongoing overarching arcs and 'case of the week' episodic elements within the bigger narrative(s). Click Here to see a decent breakdown of just how complex a single episode the Kings' series can be and how it is seamlessly able to mix that romantic melodrama into it's legal and political plots and physical scapes.

When the CBS legal drama/political soap opera hybrid debuted in 2009, it was easy to identify the glaring similarities between the show and the still-fresh details of the Spitzer prostitution scandal that was unveiled the previous spring. The series launched with an all-too-familiar scene: Alicia Florrick (played with elegant ferocity by Julianna Margulies) stands stunned and stonefaced next to her politico husband (Chris Noth)—who had just been caught cheating on her with a hooker—at his press conference announcing his resignation. It was as if Margulies had taken care to recreate, second-by-second, the expression worn by Silda Wall Spitzer as she stood by her husband in the press conference that served as inspiration for the series.

Grey's Anatomy - Much truer to American day-time saops, Shanda Rhimes "medical drama" is much closer to soaps like General Hospital, as interpersonal relationships, especially romantic are always on display being the series' corner stone and like those often long-running television soaps of the 80's, Grey's is going on season 16! However, even though a lot of what's portrayed on the series, is medically inaccurate, two doctors are executive producers and the cases themselves are often taken seriously, which is most likley why, along with the heavy serialization, Grey's is considered more of a "drama", despite those interpersonal romantic/family elements.

By any standards, Grey's Anatomy has been successful television, ranking highly in the ratings for nine seasons and entering the cultural lexicon via phrases as cloying yet catchy as 'McDreamy', the show has had its periods of being intensely irritating, and it has had its periods when it seems as if Shonda Rhimes has taken leave of her faculties, but it's also got an amazingly high batting average, particularly with every solid season that passes along in this second act of its run." The site lauded the show saying, "On average, it's been very good TV, filled with interesting, driven characters who run the gamut of professions within the show's hospital setting. It's been, by turns, a good soap, a good romantic comedy, a good medical drama, and a good interpersonal show about an unexpected workplace family. --Samantha Highfill, Entertainment Weekly

Downton Abbey - Mostly seen as a British Historical Period Drama, really has a lot of melodrama coming from both it's upstairs/downstairs approach, along with racial and class divides providing for some juicy melodramatic romances such as Sybil & Tom, Mary & Matthew, or Rose & Jack, especially since all three tragically come to an end.

Downton Abbey has been a commercial success and received general acclaim from critics, although some criticize it as superficial, melodramatic or unrealistic. Others defend these qualities as the reason for the show's appeal. David Kamp of Vanity Fair wrote, "Melodrama is an uncool thing to trade in these days, but then, that's precisely why Downton Abbey is so pleasurable. In its clear delineation between the goodies and the baddies, in its regulated dosages of highs and lows, the show is welcome counter-programming to the slow-burning despair and moral ambiguity of most quality drama on television right now."

So my own conclusion is that what Soap Operas originally were have been discarded or have evolved into varied categories either through a false perception given to the name/definition of American day-time television melodramatic shows that were "set-up" structurally like British soaps, and often visually contrasted the higher brow visuals of Hollywood (not unlike "Reality TV", I might add), which in itself also changed when someone tried to take the melodrama and interpersonal relationships aspects of those day-time soaps and apply them to a "drama" creating something like Dallas or it's spin off, Knotts Landing allowing them to have "soap operas" on prime time television and then these elements (structure, day in the life, melodrama) have now worked their way into other genres as well...