Rodan also bears a close resemblance to the Firebird in Fantasia 2000.
The scene where Ghidorah regenerates his missing head also brings to mind the Hydra from Hercules, membranous sac and all. Thank goodness he only regrows one head.
Rick and Morty stickers are on Madison's laptop cover. Word of God says that Rick Stanton is modeled on Rick Sanchez, too.
Stanton sarcastically refers to King Ghidorah as "Moe, Larry and Curley". Ghidorah's middle head fittingly bosses around the other two heads hitting them (or at least Left-head) into submission just as Moe does to Larry (Right-head) and Curley (Left-head).
A newspaper headline during the credits speculates on possibly using Kaiju feces as fertilizer, an idea previously discussed by Hannibal Chau in Pacific Rim.
The scene where Rodan swallows an ejecting pilot is shot remarkably similarly to the scene in The Giant Claw where the titular monster does the same thing to the parachuting pilot of a downed plane.
The stinger's Sequel Hook contains a reference to Jay-Z and Kanye West's "No Church in the Wild" — a newspaper headline reads "What's a King to a God?"
There's a definite influence from the Cthulhu Mythos, with Monarch discovering the existence of an ancient Titan-worshipping civilization predating recorded history, and the team expressing horror at how unnatural the Titans and their abilities are following Rodan's awakening. Godzilla's lair is a sunken prehistoric cyclopean city, and King Ghidorah is discovered in Antarctica and its powers are described as being eldritch. The redacted text in the film's credits also lists the "Old Ones" as an alternative name for the Titans. Fitting because like the Great Old Ones, Godzilla is a mighty aquatic monster while King Ghidorah is an alien from outer space. Like the Cthulhu Mythos, the movie also emphasizes that the Titans/Old Ones once ruled the Earth, and that there's not much we can do to stop them from taking it back; the difference is that here, harmonious coexistence seems to be possible.
In the novelization, one of Jonah's Mooks is nicknamed "the Mountain" by Madison. Made all the more fitting by the fact the Mook in question is large and silent.
A group of heroes traveling around and encountering monsters on a vehicle called the Argo. Soundfamiliar?