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We DO use the unit Newton for weight, but I assume you mean in common every day use.
The word "Weight", in common usage, means the scientific concept of mass. The two are interchangeable in a (roughly) constant gravitational field (the surface of the Earth), as they are directly proportional. That's really all it comes down to.
Yes, scientifically it is incredibly sloppy. When people talk about on the moon, they say "you weigh 1/6 the amount you weigh on earth". This is scientifically correct. You mass cannot change. But they will still measure this weight in kg.
I think it's just a cultural hangover. In science the two used to be interchangeable until Newton's work on gravitational attraction. This continued in common use, while the new branch of gravitational science made sure to differentiate between the two.
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