All There in the Manual: Marvel vs. Capcom 1-3do share continuity, but you wouldn't know it from playing them. In summary, a rift opens between the Marvel and Capcom worlds after Professor Xavier becomes Onslaught. The heroes eventually return to their respective worlds, but are periodically called upon again to fight bigger threats.
Alternate Universe: Many believe that the Marvel vs. Capcom games are Fake Crossovers, but on the contrary, Marvel acknowledged the continuity of these crossovers (alongside X-Men: Children of the Atom and Marvel Super Heroes) enough to assign them to their own official Marvel universe called "Earth-96169".
Attack of the 50-Foot Whatever: Apocalypse in X-Men vs. Street Fighter and Marvel Super Heroes vs. Street Fighter, Onslaught in the first Marvel vs. Capcom, two of Abyss's forms in Marvel vs. Capcom 2 (first and third), and taken Up to Eleven in Marvel vs. Capcom 3 with Galactus as the end boss.
Blocking Stops All Damage: Iceman takes no blocking damage, which means he is an ideal choice to fight the otherwise unstoppable gamebreaker Cable.
Dante: The demon slayer is clearly his teenaged self from DMC3, with powers and abilities only from that adventure, but he seems to know Trish and his foe Mundus quite well, characters that only his a-decade-older self should've known.
Magneto: Capcom pretty much gathered every single power (except Psychic Powers) Magneto has ever had since his debut, so yeah, in the hands of some writters the Master of Magnetism indeed was capable of creating Force Fields that were not limited to deflecting just metal; attacking with Beams and Energy Blasts that were said to be just a Second Mutation of his already powerful control over magnetism. Simply put, Magneto isn't shown to have all these powers in the same continuity.
Mega Man: The classic Blue Bomber has his set of copied powers from other Robot Masters (namely the Leaf Shield from 2 and Tornado Hold from 8), as well as the Mega Ball from 8 and the Mega Upper (seen in a secret, Street Fighter-style mini-game in 7 and Mega Man 2: The Power Fighters). Interestingly, he also uses the Rush Drill, a form of Rush that was intended to have been included in Mega Man 3.
Ryu: Basically, Ryu was all of his incarnations into one up to Marvel Vs Capcom 2; placing in order, Ryu's appearance is that of his young self from Street Fighter Alpha: white bandana, but his mannerisms (from his quotes) are that his older self from Street Fighter II and beyond, were he should have Ken's red hair ribbon as a bandana to simbolize Ryu's maturity and confidence in his strength while he was troubled with it for his struggle with the Satsui no Hadou in Alpha (he wears it in the first MvC, even though the animations are still the same as before), and he seems to know Sean from Street Fighter III quite well to offer training sessions in his ending for Marvel Vs Capcom Clash Of The Superheroes, a character that Ryu will only meet half-a-decade later. Marvel vs. Capcom 3 came and returned Ryu's red bandana (MvC2 snapped back to his Alpha incarnation) and portrayed him as the full-fledged 30-something everyone came to know him as (his MvC3 model derives from the Street Fighter IV one), pretty much fixing all the issues since those were the only notable changes in Ryu's character during his journey.
She-Hulk: Her costume and assertive personality is that of when Jennifer became an established character, but she seems to have some of her early Leaning on the Fourth Wall characteristics, from the time she didn't have a main uniform and was more irresponsible than playful.
She even lampshades this when beating Deadpool:
"You know, if this game was made in 1991*
This was the year when Deadpool was created, and his current persona was established way later
, I'd be the one whacking YOU with a health bar."
Zero: While Capcom characters keep their power set mostly reserved from that one game they appear in, Zero's abilities are all over the place, it is a mix of pretty much X4 to X8 (i.e. at least one special technique from every main game he's been playable in).
Immune To Flinching: larger characters like Juggernaut or Sentinel have various amounts of Super Armor — there is also "Hyper Armor" status which temporarily renders a character completely immune to flinching and knockback. Boss characters (like Galactus) have this as their default state.
Leitmotif: Save for 2, all games have featured this for every playable character. Out of all of them, Captain America and Spider-Man hold the distinction of having the same theme for every game they have been playable in, as well as Sentinel, Gambit, Blackheart and Shuma-Gorath.
Marth Debuted in Smash Bros.: As Marvel Comics' namesake medium (comics) were never popular in Japan (the majority of them were never published there, either), this is naturally how people in said nation feel about several Marvel characters when they appear in any Marvel-licensed game made by Capcom.
Non Dubbed Grunts: In the Marvel vs. Capcom games, sound clips are often reused from older games.
Magneto (only absent in Marvel Super Heroes vs. Street Fighter) and Shuma-Gorath (only absent in Marvel vs. Capcom) are next to them.
Morrigan appeared in all the games, aside of the Marvel-only games and the Street Fighter crossovers.
In the same vein as Morrigan, Strider Hiryu only missed the original MvC3.
Another example is Akuma, who has the distinction of being the first Capcom character to appear in the series, as he was a Secret Character in X-Men: Children of the Atom. He was also a Secret Character in X-Men vs. Street Fighter and Marvel Super Heroes vs. Street Fighter, appeared somewhat in Marvel vs. Capcom as a moveset for Ryu, and was a regular in Marvel vs. Capcom 2, returning to his unlockable status in 3.
And Zangief, who appeared in every Street Fighter/Capcom crossover with Marvel, except in 3, where he's replaced by Haggar.