Stormbreaker should have been erased from existence in Avengers: Infinity War because of its connection to Groot, but there's a good reason why it survived the snap. Groot, along with several other members of the Guardians of the Galaxy, was killed when Thanos gathered the Infinity Stones and snapped his fingers, right after Thor failed to kill him with Stormbreaker. The only Guardians who were left standing after the snap were Rocket and Nebula. Groot didn't return until Professor Hulk used the Stones to bring everyone back near the end of Avengers: Endgame.
In Thor: Ragnarok, Thor lost his greatest weapon when Hela destroyed Mjolnir just by crushing it in her hand. The God of Thunder remained unarmed for the remainder of the film, although he was still able to use his lightning powers. He didn't find a replacement for Mjolnir until Avengers: Infinity War, when he, Groot, and Rocket set out on a mission to build a weapon that he could use to kill Thanos. With the help of the dwarf Eitri, Thor forged Stormbreaker, an incredibly powerful axe based on the weapon wielded by Beta Ray Bill in Marvel Comics. Much like with Mjolnir, Thor is able to channel vast amounts of lightning through the axe.
The axe nearly didn't get finished. however, and Groot had to cut off his arm so that it could be fashioned to the axe blade. Groot's arm became the handle of Stormbreaker, making him instrumental in the origin of an important weapon from the Marvel Universe. But despite Groot's involvement in the creation of Stormbreaker, his sudden death in Infinity War doesn't have any impact on the axe. Groot disappears with the snap, but Stormbreaker remains in one piece rather than disappearing with him.
Here's why Stormbreaker wasn't affected by the snap. When Groot cut off his arm, it became like any other severed limb and ceased being a part of him. When Groot sacrificed himself to save his friends in Guardians of the Galaxy, pieces of him were left behind. They were no longer a part of Groot. The same is true with his arm, which is now just an axe handle, and thus doesn't disappear from existence when Groot does.
However, this fact doesn't diminish the role Groot plays in a pivotal moment in the movie. Groot is still the reason why the axe is able to be completed. While it unfortunately doesn't serve its intended purpose in Infinity War -- which was killing Thanos -- Stormbreaker is the weapon that ultimately kills the Mad Titan in the beginning of Endgame, and it factors heavily into the movie's final battle when it's wielded by both Thor and Captain America.
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