At the end of Avengers: Infinity War, we have seen Groot been erased due to Thanos' snap.
So shouldn't the handle he made for Stormbreaker get erased as well?
Just wanted to know if there is any story plot behind the snap I am missing.
Thanos' "snap" erased half of all LIFE in the universe.
Since Groot was alive he was one of those erased as part of the snap.
BUT, the handle of Stormbreaker is made from the "dead" (sacrificed) arm of Groot ...so it's not alive and so not affected by the Infinity Snap.
The snap does not remove a life's history from the universe, merely stops them from existing going forward.
Groot Jr's arm is no longer part of him. He cannot make it grow, sprout flowers, change it's form etc. It cannot say, "I am Groot". So when Groot Jr. is turned to ash, there is no reason for something no longer part of him to be effected.
The MCU writers made it pretty clear that only things that were on a person's body at the time they turned to ash were destroyed. For example, in the post-credits, Maria Hill gets turned completely to ash, including her clothing and whatever may have been in her pockets, but Nick Fury is able to grab a pager, call out an emergency code, and then dropped the pager before the ash reached it, and it survives the snap, even though nothing else of Nick's personal effects survived.
A helicopter crashed into a building, and a car crashed in front of Nick and Maria, demonstrating that it doesn't affect things that the person was touching, just things that could be considered "on their person" at the time of death. Since the handle is nowhere near Groot Jr, it's not "on his person" and wasn't affected. Also, it is hinted that a cutting of Groot is not the same as the original, much like you are a separate person than your mother or father. Even if you submit that Stormbreaker's handle was alive, there's no evidence that it is the same living creature that is Groot Jr.
Either way, the handle had an equal 50% chance of surviving even if it was alive, and that 50% chance would have been independent of Groot Jr's own chance of survival. Odds are, if it were alive, this information was withheld by the writers as a way to possibly revive Groot again in the next movie, even if the heroes never figure out how to recover from the snap.