The game's ranks use the same naming system as recurring equipment sets in the series, and mostly go in the same order of power — Bronze > Silver > Gold > Platinum > Mythril > Diamond > Adamant > Crystal.
Sephiroth's introduction video shows a recreation of his post-final boss duel in Cloud's subconscious, before transitioning to Midgar in flames, just as when Meteor wreaks havoc upon the city at the ending of their home game.
Rinoa's introductory video begins with a recreation of the flowerbed field scene at the opening video of her home game.
Story mode
The events of the first Final Fantasy happened between the first Dissidia and NT; when the Warrior of Light finds the Cornelia Field arena, he remembers this is where he began his quest to save Princess Sarah from Garland. Cue Garland's arrival shortly thereafter, commenting on how it seems they are destined to fight, even without Stable Time Loops bringing them together.
Whenever Cloud and Sephiroth cross swords, streams of light fly from the connection, as they did in Advent Children and Kingdom Hearts II.
Zidane "never thought that Lightning was the motherly type," but in her home games, she has shown this side from time to time. In her and Serah's backstory, their parents died when they were young, forcing the older Farron to grow up early to take care of the younger. During the first game, she takes care of Hope Estheim and helps him grow.
On meeting Lightning and learning the situation she's ended up in, Snow's first assumption is that she's intent on kicking his ass. Afterwards, it's revealed that it was a ruse to make sure that her intentions were as well-intentioned as he remembered. Are we talking about Snow holding Odin in Dissidia NT, or the Patron-vs-Savior fight in the opening of Lightning Returns?
Ramza and Ace made cameo appearances during the credits; Ramza appears with Cloud, Squall, and Zidane, and Ace appears with Vaan and Y'shtola. This lines up with the release of their respective games - Tactics was released shortly after VII, and Type-0 came out between the releases of XIII and XIII-2 (Lightning appearing at the start of the montage means she's absent later on).
Bartz and Exdeath first appear in story mode in the Interdimensional Rift, and Exdeath's dialogue is a word-for-word quote of his lines when confronted at the end of Final Fantasy V. This reveals that the two have been pulled back into the cycles in the middle of their final confrontation in their original game, which is why they're surprised to see Cloud and Bartz takes a moment to realize what's going on.
Terra and Zidane traveling together is an allusion to both of them sharing the same Trance transformation in their original games. Kuja even joins later and they enter Trance together before a battle. It also references Terra's travels with Locke, another thief, and Zidane flirts with her a bit similar to how Edgar did when meeting her.
Additionally, the birth place of Zidane and Kuja is called Terra.
The summons allow for several:
Terra's bond with Espers in her home game comes into play here, allowing her to sense the summons of this other world. For that matter she's used to the Espers being friendly, and is taken aback when the summons of this world attack on sight and don't listen to reason.
When Terra and Zidane find a summon, it's Ramuh, who just happened to play an important story role in both of their home games.
Immediately after, Lightning and Squall encounter Odin, who was Lightning's personal Eidolon in XIII and had a bonus story bit if Squall found him soon enough in VIII.
The summon found in the Alexandria Square arena is Alexander, who appears over the castle and proceeds to destroy the city, much as Bahamut did in the Final Fantasy IX story event where Alexander was summoned to fend him off. Kuja, who knew what summon he was looking for, lampshades the obviousness of where it would be found, and Zidane cracks that it isn't as friendly as the version he knows from back home.
Meanwhile, the summon fought by the Warrior, Cecil, Noctis, and Garland is Bahamut, who was in Final Fantasy I before he was ever a summoned creature, played an important story role in XV, and had a minor supporting role in IV.
Firion and Tidus are accompanied by Jecht for their summon fight. Leviathan, similar to Bahamut, had a role in II before summons were introduced to the series, and the water-elemental summon is thematically appropriate for a pair of Blitzball players.
Shinryu's first move is always Tidal Wave, his Signature Move.
Boss battles
Vs Ifrit
Bartz tells Ifrit he "[doesn't] look too tough", referencing his status as a mid-boss for an early game dungeon in V, and beating him to get his summon was obligatory.
Sephiroth says "one swipe is all I need", a reference to Ifrit's appearance in Crisis Core and Sephiroth beat him in a Single-Stroke Battle.
Rinoa says "if I win this fight, maybe I can be a SeeD too!" Defeating Ifrit is part of the graduation exam for SeeD cadets.
Y'shtola says "I will not let you shed blood in this world." Ifrit stands out as an Axe-CrazyKneel Before Zod type in XIV.
Ardyn snarkily comments to Ifrit "You'll soon be back in my employ!" Ardyn would infect his world's Ifrit with the Starscourge in Episode Ardyn, making the Astral his puppet.
Vs Shiva
Firion sounds wary as he remarks "you will not provoke me", a reference to Shiva being The Tease towards him in the PSP Dissidia games, which in turn was a reference to the infamous moment in II when Firion was almost awkwardly seduced by the Lamia Queen.
Golbez declares "my dragon's bite is much colder", a reference to his use of Binding Cold before summoning the Shadow Dragon in IV.
Bartz says "three-on-one's fair enough!". In V, Shiva entered battle with three Ice Commanders, making it an even match against your four party members.
Kefka demands that Shiva not "make [him] throw [her] away like the last one!" An unpleasant scene in the Magitek Factory in VI had the jester tossing Ifrit and Shiva into a garbage chute.
Squall warns her "I know every move you'll make." Shiva was one of the starting Guardian Forces in VIII, so every ability she had was acquired while in Squall's possession.
Jecht asks Shiva to "Sing us a hymn, why don'tcha", referencing the Hymn of the Fayth from X. Additionally, Shiva is indirectly responsible for the heroes learning the Hymn of the Fayth can still reach Jecht as Sin, as it was near her temple that Sin was stopped by the song.
Kam'lanaut says "join me and tragedy shall not befall you". Shiva's backstory in XI claims she was a young princess who was overthrown and banished to the northlands where she froze to death.
Lightning asks Shiva if the summon "think[s she] can take [the champions] alone", which references Shiva being two entities in XIII. When Shiva Turns Red, she does indeed split into Twin Sisters.
Y'shtola says "So long as your soul's untethered!" This makes two references; one to her companion Ysale who could manifest Shiva and pulled a willing Heroic Sacrifice, while the original Shiva willingly let herself be eaten body and soul by her dragon lover Hraesvelgr, also as a Heroic Sacrifice, though a more indirect one.
Noctis says "hate to do this, but I have to," referring to Shiva being the most helpful and benevolent of the Astrals in XV, as well as a friend of Luna's in her alter-ego Gentiana.
Ardyn says "try to keep your hands to yourself this time", referring to when Shiva froze him solid by planting a finger on his lips to shush him.
Vs Ramuh
Locke says "I think we'll get along just fine". Ramuh was perfectly friendly and hospitable to the party in VI.
Tidus proclaims "I can dodge lightning all day!" referring to the infamous sidequest in X to dodge 200 bolts in the Thunder Plains, which can take players a very long time.
Snow claims that "not even lightning can stop me" when facing Ramuh, a rather cheeky reference to Lightning's unsuccessful attempt to talk him down from absorbing the Chaos of Yusnaan into his body.
Noctis expresses relief that he "didn't have to search for runestones" before fighting Ramuh. In XV (and Episode Duscae), Noctis received Ramuh's power after such a search in the less-than-navigable Duscaean landscape.
Y'shtola says "I assume there is no treating with you?" Ramuh is a Token Good Teammate among primals and negotiation is actually a valid tactic.
Ardyn snarks "I remember you being a bit...bigger". Astrals in XV were Kaiju sized, and Ramuh could hold Noctis in the palm of his hand.
Vs Leviathan
Y'shtola says "We will not go the way of the drowned." Primals can cause normal people to become Brainwashed and Crazy thralls, Leviathan's branded slaves being known as The Drowned.
Noctis asks "how about I enter into a covenant with you too?" He did in XV, but it took a lot of convincing (and a boss battle) to get Leviathan to agree to it.
Ardyn says "awaken from your slumber, have you?" referring to Leviathan in XV sleeping under Altissia until Lunafreya woke it up to form a covenant.
Vs Bahamut
Bartz tells Bahamut "You lose, you serve", referencing the plot causing him to appear and challenge the heroes; while fighting him was optional, the party earned his summon upon victory.
Terra remarks that she "didn't have to fly to find" Bahamut, whose magicite was claimed by defeating the sky-wandering Deathgaze in VI.
Kuja says "o, had you but bowed your head in supplication", referring to how he forcibly took control of Bahamut in IX.
Vayne remarks "no fortress in impregnable", an allusion to his Sky Fortress in XII bearing Bahamut's name.
Noctis remarks "I'm more of a king than you'll ever be!" in a particularly angry tone. The Bahamut of his world was not only a god-king presiding over Eos, but an Omniscient Morality License carrier with very little in the way of sympathy, and was the final antagonist in the Alternate Timeline of the Dawn of the Future novel (detailing the cancelled DLC episodes).
Ardyn responds to Bahamut's presence with "I never much cared for you" in an understated, irritated tone. Given that Bahamut forced him into the role of the Immortal Cursed as revealed in Episode Ardyn, he has good reason to dislike any of Bahamut's incarnations.
Y'shtola says "You will cause no calamity here!" A defining moment in the game's history is when Bahamut causes a continent-wide Apocalypse How after being freed.
Vs Odin:
Locke asks if Odin "always [has] such a stony gaze", referencing Odin being petrified when the party found him in VI.
Kefka claims "the spell is mightier than the sword", referring to how Odin in VI was defeated when a sorcerer was immune to Zantetsuken and petrified him.
Yuna asks Odin "How much will it cost". Odin was absent from X; the local Diagonal Cutter was Yojimbo, who demanded payment to assist her.
Kam'lanaut asks "are you the lord of this underworld?" Odin in XI was the summon of Darkness and dwelled in an underworld in the depths of Vana'diel.
Lightning declares "I refuse to roll over and die!" Odin in XIII was her eidolon, who manifest when l'Cie lose their resolve and appear to either kill them or be subdued as proof they can keep going.
Y'shtola says "'Twas almost effortless to find you." In XIV Odin is a wandering raid boss that spawns very rarely and in semi-random locations, causing a rush to find and fight him.
Zenos comments that "Your cowardice is unbecoming.", again referencing Odin's status as a wandering raid boss.
Vs Alexander
Kain says "I see there are giants here, too", referring to Alexander's design being based on the Giant of Babil.
Locke hopes that Alexander, looking like a castle, "doesn't go underground" in reference to VI's Castle Figaro.
Rinoa says "I've drawn you out of hiding!" Alexander had to be Drawn from Edea in VIII to get it as a GF.
Zidane's reaction to fighting Alexander is "May our pleas for help reach you!", which paraphrases Garnet and Eiko's calling of Alexander in IX.
Kam'lanaut says "here, you're nothing but an iron giant". Alexander in XI was the Avatar of Light and one of the highest-ranking Avatars, and its nickname was "the Iron Colossus".
Snow proclaims himself "too stubborn to die" when facing Alexander, words that Lightning used to describe him in XIII.
Ace proclaims that "The power of the Verboten will be ours!" when facing Alexander, who was indeed a Verboten Eidolon in Type-0.
Y'shtola says "The resemblance is uncanny..." Y'shtola was present through most of the Alexander storyline. She also doesn't have any harsh words for him as the end of Alexander's story reveals he was Good All Along.
Vs Shinryu
Yuna says "I will end the spiral of sadness!", a direct allusion to ending the cycles of Sin's rebirth in X.
Y'shtola proclaims "A savage beast deserves a savage end!" Shinryu in XIV is a mindless berserker. It may also be referencing Zenos Yae Galvus, the villain of Stormblood who is described as a depraved beast even by his own allies and manages to Grand Theft Me that version of Shinryu.
Zenos quips that "This dragon, too, shall be made to kneel!" Zenos acts as the final MSQ boss of Stormblood by using his Resonance to possess Shinryu.
