https://www.ign.com/articles/thunderbolts-ending-new-mcu-team-new-avengers-doomsdayThis IGN article explains it well, but for those that want some comics lore without reading a whole article, here we go:
In ~2005, the Avengers had disbanded but a new group of heroes came together to stop a breakout at the Raft prison. New Avengers was the name of the comic.
After Civil War, you had the "official" Mighty Avengers (Tony Stark's government-registered team), which included the Sentry, and then you had the "rogue" Team Cap Avengers who were underground (Wolverine, Luke Cage, Iron Fist, Spider-Man, Doctor Strange, Spider-Woman). It kicked ass.
Eventually, Norman Osborne took over Shield and therefore had the official Avengers team under his thumb. He ended up stealthily replacing its members with his cronies. Bullseye, for example, was the "new Hawkeye". Norman Osborn was Iron Patriot (a Cap/Iron Man mix). They were bad dudes mostly doing bad things, or taking out other villains to consolidate their own power.
So there is precedent for two Avengers teams. I don't think Bucky vs. Sam will be a very big part of Doomsday, if it is at all, but the Thunderbolts could be the government stooge Avengers (technically still under Val's jurisdiction?) and Sam's could be your true inheritors to the title. Or it could be the other way around - Sam's team is your media darlings and the "New Avengers" are your B-list bad boys who are willing to do the dirty jobs? I guess we'll see.