They are the same company, similar to how Gap and BR are the same company. I wouldn't compare Madewell to Gap or BR really though, since they carry a lot of non-madewell brands as well.
They are owned by the same co. but not really at all the same. Madewell focuses more on denim and is definitely more casual and has a younger target. I will say though, Madewell's denim is more flattering on me than J.Crew's because the back pockets aren't as huge.
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They are the same company, similar to how Gap and BR are the same company. I wouldn't compare Madewell to Gap or BR really though, since they carry a lot of non-madewell brands as well.