His real name, Johann Schmidt, is a coincidence because that's the character's name in the comics, but it sort of counts; "Schmidt" is the German form of "Smith".
His introductory scene has him approaching in a car, and then walking into the church hideaway with his feet being focused on first, and Erskine's explaining about Red Skull has the first flashback depicting him with several copies of himself, similar to a certain Agent in The Matrix. The shot of a dozen Schmidts is a particularly strong example as it makes absolutely no sense except as an Actor Allusion.
His fight with Cap in a plummeting plane, which makes a suspended-gravity environment, is reminiscent of a scene in The Matrix.
At a meta level, things like Red Skull's fighting style and even how he levels his gun ("Unfortunately I am on a tight schedule...") have apparently stayed with Hugo Weaving after the Training from Hell he went through to play Agent Smith.
Hugo Weaving:"It was fun to play; I enjoyed the outrageousness of the German accent that I employed and I enjoyed the extraordinary mask and costume, even though it was unbelievably hot inside it. I enjoy mask work; I enjoy trying to animate masks and reveal certain things that the mask itself might not reveal. V for Vendetta was another example of that, but there was less animation within that mask versus the Red Skull. With Marvel, it’s pretty basic stuff: accept the deal and enjoy the ride. It’s not a major stretch for an actor, but on the other hand the difficulty with the Marvel universe is maintaining a link to a human dimension within such an extraordinary, technological CGI universe. In terms of me going back and doing another one, I don’t know. I’m not sure what they’re up to with the Red Skull."
Dawson Casting: Inverted with Stanley Tucci being casted as Erskine. Tucci was in his early fifties while character who's supposedly in their seventies.
Was actually deliberately Averted in the case of Captain America himself (Sebastian Stan originally auditioned for Steve). The crew felt that it was necessary for the Cap to be played by a genuine American.
Thoroughly averted in the case of Neal McDonough as Dum-Dum Dugan, who shares the same nationality and home city of Boston, MA, as well as Irish lineage, as well as Bruno Ricci (who is actually French) as Jacques Dernier.
English Natalie Dormer as the American soldier who kisses the Captain.
Korean American Kenneth Choi as Japanese American Jim Morita.
English Lex Shrapnel as the American soldier Gilmore Hodge.
Fan Nickname: Pre-serum Steve is often referred to as 萌やし (moyashi) or "beansprout" in East Asian fandom due to his small size.
I Knew It!: As confirmed by the sequel, Bucky survives and becomes the Winter Soldier.
Money, Dear Boy: Hugo Weaving has gone on record stating that he is indifferent to the film and has very little interest in reprising his role as the Red Skull, due to the difficulty of the makeup process involved. However, as seen above under Awesome, Dear Boy, his attitude towards it has changed, potentially opening the door for his return.
The Other Darrin: Howard Stark was portrayed by Gerard Sanders in still photos, John Slattery in archive footage and Dominic Cooper in life. Justified as all three depictions are from different time periods — the still photos are from around Tony Stark's pre-teens and beyond, the archive footage from his childhood, and in real life it's the '40s long before he's even born, so Howard ages significantly between each depiction. Slattery and Cooper have continued to portray the character in subsequent MCU works depending on what the time period depicted is.
Sebastian Stan originally auditioned for the role of Captain America, but was given the role of Bucky Barnes.
Chris Evans turned down the offer to play Captain America several times; not out of dislike for the role, but because he feared what the effects of the sudden increase of fame would be on his private life. Then Robert Downey, Jr. convinced him to take the part and thus gain the freedom to sign on any other role he'd want to after, and the rest is history.
Up until a very late stage in pre-production, HYDRA would have been a blatantly Nazi paramilitary organization, with swastikas on their uniform. The implication is still present in the final version, however. In addition, deleted scenes also had HYDRA explicitly attacking Nazis in addition to Allied powers.
Emily Blunt was offered the role of Peggy Carter and her husband John Krasinski was heavily considered for Steve Rogers. The story goes, though, that John turned down the role after suiting up for a test run, declaring "I am NOT Captain America".