Though they are often compared to DC's Suicide Squad, the Thunderbolts have been many different things over the years, but their complex origin story hasn't been replicated since they arrived.
Although they are a superhero team of villains, the original Thunderbolts were much more complex and a completely different concept than DC's Suicide Squad. The team was created by Kurt Busiek and Mark Bagley. Their first appearance was in 1996's Incredible Hulk #449.
The Thunderbolts vary greatly from the Avengers as they are not strictly speaking an institution. The teams have come and gone over the years with no clear path linking one incarnation to the next. Sometimes they take on a Suicide Squad-like mission, enlisting incarcerated villains to do good. Other times they are simply a group of hardened heroes brought together by happenstance, such as in 2012 when Red Hulk formed a team.
Eventually, Zemo’s teammates had a change of heart. They enjoyed being heroes and, rather than turn on the Avengers, decided to be heroes for real. Of course, Zemo didn't like this plan and clashed with them. But the remaining Thunderbolts found a new teammate in Clint Barton, aka Hawkeye. Barton had a criminal past, so he found companionship in the group of former villains.
It is unlikely that Marvel will have a team quite like the original Thunderbolts anytime soon. For one, they can't copy the idea exactly. Writers would have to come up with some unique spin on the premise, such as having superheroes create new monikers for a villain team instead. But super villain teams have largely taken a backseat in the greater Marvel universe.