"Fighting the Chitauri is nothing compared to keeping track of all of my goddamn receipts."
Team-up movies don't necessarily need a SHIELD. In the original King Kong vs. Godzilla, the monsters fought each other because they happened to be in the same time zone. In the original run of Universal Monster films, Frankenstein and his peers usually got rowdy because some a*****e scientist decided that it would be a good idea to implant the brain of a ghost into the body of a vampire or something, and there's nothing that monsters hate more than malpractice. It's silly, but it worked.
"I will install the anus of the Phantom of the Opera into the hands of a Frankenstein ... FOR SCIENCE."
"Not so!" huffed Hollywood, putting its cigar out in the open eye of a screenwriter. "People wouldn't buy that kind of absurd logic in our stories about the reanimated corpses of Egyptian supergods. We have to create our own secret government organization that gets these guys together." Thus, Project Monarch and Prodigium were born, with little regard for what their actual purposes in the movies would be.
Both organizations are around to keep track of monsters and ... make sure they're cool? Project Monarch appears in Godzilla and Kong: Skull Island, but all we really learn about them in both of those movies is that their members like to stand around while the other characters participate in the plot. They say things like "Let them fight," and while that's a rad line to say while nonchalantly looking off into the distance, who the f**k are you, dude? You haven't helped at all in this whole film, and now you get to be the guy to decide whether or not it's cool for a trio of 400-foot-tall mutants to destroy a major city while they hopefully murder each other?
Warner Bros. Pictures