In the MCU, the DC characters would have to be fictional. Marvel doesn’t have the film rights to all of its own characters, much less the DC characters. I don’t recall any references to Marvel characters in the DCEU movies, but in the DC/CW series The Flash, characters occasionally mention Marvel characters and movies as fictional, pop culture references.
In the comics, there were quite a few aborted attempts at crossovers. Back in the 70s, there were a handful of cross-overs that actually got published. They were one-offs; in that specifici issue, the characters were portrayed as existing in the same world and as always having been, they just hadn’t run into each other before. Outside of those one-off issues, though, those encounters were never directly referenced, and the characters existed in totally separate continuities. Writers and artists would occasionally slip in oblique references as in-jokes, but nothing official.
There were also analogs of the other company’s characters that would appear from time to time. Marvel’s Squadron Supreme, for example, were a team of heroes from an alternate universe that were remarkably similar to the then current line-up of DC’s Justice League (the story is, the writer of Avengers at the time tried to arrange an official cross-over, couldn’t get it approved, and so he just wrote the story anyway, with thinly disguised analogs).
Marvel and DC have done a few official, in-continuity cross-overs. In the 90s, they co-published the Marvel vs. DC limited series and the Amalgam comics event (where both companies published “amalgam” comics that combined aspects of their characters, like Dr. Strangefate who combined elements of Dr. Strange and Dr. Fate). That series introduced a new character, Access, who’s powers were supposed to let him travel between the DC and Marvel multiverses, and he was co-owned by both companies. He’s only had a handful of appearances since then, though, and I don’t think he’s even referenced the other company’s characters in those.
In 2003-4, they co-produced the JLA/Avengers limited series, where the JLA and the Avengers met each other in what at the time was supposed to be an official in-continuity event for both titles. I think they’ve had a couple of other more recent cross-overs that were similarly supposed to be in-continuity.
In those cases, the Marvel universe and the DC universe were portrayed as separate universes, even separate multiverses, and the cross-over was in-continuity a cross-over between different universes.
So, long story short, the DC and Marvel characters potentially exist in parallel multiverses and can cross over, but officially they don’t co-exist in the same world.