Is there any idiom in English likening a person reaching adulthood and moving out of his/her parents' house to a bird leaving the nest made by its parent birds? If not, what idiom in English would commonly describe a young adult moving out of his/her childhood house (the house where he/she was born and grew up)?
A grown-up child leaving the parental home is sometimes said to 'fly the nest'. The initial loneliness sometimes felt by parents after their children have left can be called 'empty nest syndrome'.
fly the nest
When children fly the nest, they leave their parents' home to live on their own.
When their children had flown the nest, they moved to a cottage in Dorset.