In Episode 16, she can be heard humming "Invoke", SEED's first opening theme, while packing her bags.
In Episode 23, Mao's master, whose favorite Gundam girl is Murrue Ramius, can be seen ogling Rinko.
Kirara's idol outfit is white and has pink hair which is tied in Girlish Pigtails, not unlike Madoka Kaname, who is also voiced by Aoi Yuuki. (It is unknown what is Kirara's natural hair color, but Madoka is naturally pink haired for sure.)
As Mihoshi, she ties her hair in a very long ponytail with two short sidelocks, wears a thick clothing and is also a pilot (though she joins in Gunpla Battles as Kirara, however), not unlike Akira Renbokoji, who is also voiced by her voice actress in another Sunrise anime, Valvrave the Liberator.
Sazaki is often called Gyandere, due to his obsession with Sei and his preference of Gunpla.
Rinko is called Mama-n, the suffix ~n is used in Japan as to point out cuteness aloud.
Also it sounds the same like the Japanese slang Maman, who is the equivalent there of the phrase MILF.
The moment Sengoku Astray appear fully as model kit, it earned the title Weeaboo Astray, being overtly Japanophile all around (even the name Gundam is written in Kanji)
In certain Imageboards, the show also called "Gunpla/Gundam Boy Friends" due to abundance of shipping material, both the Ho Yay and straight variety.
Rainer Cziommer, due to his Wild E. Coyote tactic, is now know as "butt-hurt NTR guy" in Japan.
The alliance of pilots who attack Fellini in episode 11 was quickly dubbed "Team NTR", since their entire motivation was to exact revenge against Fellini for stealing their girlfriends (though he insists that they came on to him).
Aila's Qubeley Papillon is slowly getting the nickname "Uberley" for being overpowered against other Gunpla that it faces.
Nils Nielsen is now known as NiNiNi - for Ninja Nils Nielsen, which is what he becomes when he's not doing Gunpla fight.
As soon as the Renato brothers started to snark at Luang Dallara's defeat during the episode 17 stinger, the western fandom quickly named them "the evil Lockon twins".
The strange friendship/Ship Tease between Reiji and Aila has been labelled as "The Meatbun Chronicles", due to that food playing a role in their first several meetings. As a result, both of them received the nicknames "Mr. and Ms./Mrs. Meatbun."
Dark Matter Exia is often jokingly called "Edgia" due to its "dark and edgy" colorscheme and all of the sharp edges incorporated into its design.
The GBF universe is also sometimes called "Gundam Valhalla" due to the numerous cameos of previously deceased Gundam characters being shown living happy and carefree lives.
"Trans-Amazing" for Amazing Exia's Trans-Am.
"Build Hell and Heaven" for Star Build Strike's Double Build Knuckle, which was used against Exia Dark Matter.
Meanwhile, due to the events of Episode 25, Aila quietly received from the fans the title "Arian's Future Queen".
Ditto also for Mobile Suit Gundam Wing: This series marks the reunion of the voice actors of two of the Gundam Doctors (Doctor J and Master O), as Master Chinnan and Mr. Ral, respectively.
Screwed by the Lawyers: Due to various legal agreements with Japanese TV networks, Mobile Suits that have aired on MBS and TBS in the last 5 years cannot be shown in Build Fighters. This means that, while suits from season 1 of Mobile Suit Gundam 00 are present and represented, suits from season 2 are a no-show; same goes with Mobile Suit Gundam AGE. AGE's case might be related to the co-creator, Level5
Although given its OVA release nature, Mobile Suit Gundam Unicorn is apparently fair game, considering a Loto did show up in the series, as did the Unicorn and Kshatriya (albeit as static models).
That said, the appearance of the Amazing Exia Repair in the final episode made fans squeal for joy... especially when they remembered that 00 Season 2 suits became fair game two days before Episode 25 aired.
Actually, the Amazing Exia is a double case of this: Careful examination reveals it's not actually an Exia, but an Exia Repair II. The staff did mention on Twitter doing something that pissed off the brass...