Why are the CW series called Arrowverse?
Is this an official calling? If it is, is it due to the series "Arrow" being the first to debut on screen?
Yes, It's because Arrow was the first.
The Flash and Supergirl showrunner Andrew Kreisberg called it as Arrow-verse,
First of all, we don’t even know if we could, legally. And quite frankly, just making the Arrow-verse, which is what we call Legends, Flash and Arrow work and then making Supergirl work, it's enough of a challenge without having to add a fourth player.
The Arrowverse is a TV universe that started with Arrow, continued with The Flash etc...
Arrowverse is an American media franchise and shared fictional universe that is centered on various television series airing on The CW and web series airing on CW Seed, developed by Greg Berlanti, Marc Guggenheim, Andrew Kreisberg, Ali Adler, Phil Klemmer, and Geoff Johns, based on characters appearing in publications by DC Comics. The shared universe, much like the DC Universe in comic books, was established by crossing over common plot elements, settings, cast, and characters.
Further on wikipedia,
The first television series was Arrow, based on the character Green Arrow, which debuted in October 2012.
The universe was expanded with the series The Flash in October 2014, which is based on Barry Allen's Flash. The universe expanded further in August 2015 with the animated web series Vixen on The CW Seed. Supergirl premiered on CBS in October 2015, before moving to The CW in 2016 for its second season.
In January 2016, Legends of Tomorrow debuted as part of the Arrowverse.
In November 2017, Ray Terrill / The Ray appeared in the crossover story "Crisis on Earth-X", a few weeks before the December 2017 introduction of the animated web series Freedom Fighters: The Ray. The animated web series Constantine: City of Demons joined the franchise in 2018. Two promotional tie-in live-action web series, Blood Rush and Chronicles of Cisco, were released in 2013 and 2016 respectively. There have also been crossovers of characters from DC Comics television shows which air on other networks to Arrowverse properties; Matt Ryan reprised his role as John Constantine from the NBC live-action series Constantine, in guest appearances in episodes of Arrow and Legends of Tomorrow, before becoming a series regular for the latter.
Also note that there are few shows that are based on DC characters but resides outside of the Arrowverse.
Lucifer (airs on FOX)
these sit firmly outside the Arrowverse.
Black Lightning (which, like the Arrowverse shows, airs on The CW), Krypton (coming to Syfy) and Titans (set to launch on DC's new streaming service).
The CW is firmly insisting that Black Lightning is not part of the Arrowverse.
Titans [...] is also being positioned as a standalone.