Bucky (Winter Soldier) has a metal arm (possibly made from vibranium?) but it's an attachment to his body as far as I understand. Why, when Thanos destroys half of the universe at the end of Infinity War, does his metallic arm also disappear with him? Shouldn't it be left behind as his gun did?
What exactly is happening at the end of Infinity War is still somewhat unclear. These people are not being edited out of history like they never existed, because we see their cars or helicopters they were driving/flying crash.
If they had never existed, people would not remember them and the vehicles they were driving or the gun they were carrying wouldn't have been there too.
So people are being removed from the Universe at that moment in time. We do see that people's clothes and personal effect also dissolve into nothing, we do not see their clothes fall to the ground after the person has been removed.
Bucky is more attached to his arm than his clothes, so it would appear that Bucky's arm falls into the category of things that are removed from the Universe along with the person. We also see Falcon's wings also get dissolved, so the power isn't just editing the person out of the universe, it is somewhat broader than that.
According to the directors it's because...
Those items are part of their identity
Avengers: Infinity War's directors Anthony Russo and Joe Russo (who also directed Captain America: The Winter Soldier, Captain America: Civil War, and Avengers 4) revealed why objects like Spider-Man's suit, Star-Lord's clothing, and other material or weapons being carried or worn by heroes disappeared with them after Thanos snapped his fingers. According to Joe Russo, the objects faded with the heroes and other victims of Thanos erasure because they are "part of their identity."
Anthony Russo chimed in: "Whatever was elemental to somebody’s presence went with them."
Whether this is logical in-universe is open to debate but out-of-universe it make sense.
Firstly this would have involved another round of complicated SFX in a movie already packed with fiddly technical requirements.
Secondly it would have looked a little messy, and possibly a little silly, on screen if leather and spandex suits, helmets, clever gadgets and all that lovely Wakandan metalwork all clanked down.
The Russo brothers do not really make a compelling argument for their vision in these scenes, but they definitely made the correct stylistic choice.
I think it's because Thanos doesn't really control how the people are removed from the Universe.
A fan theory suggest peoples are being "moved" to the Soul Stone Dimension. Since the soul stone is somehow sentient it may want to preserve the people as they see themselves: Falcon with his wings, Bucky with his arm, and Spidey with his armor.
The remaining gear is somehow not relevant to the holder personality. Fury's pager is just a device plot and Bucky's gun is a weapon.