I don’t know a family personally that has siblings with identical names. But I know of some anecdotally, and the phenomenon occasionally appears in the news or discussion forums for one reason or another. George Foreman famously has five sons named George (‘so they would always have something in common’). In my culture it’s unusual, but it happens.
Toni Morrison treats this familial anomaly with comedy and flair, albeit with non-biological siblings, in her acclaimed novel Sula (1973). In Medallion, Ohio, in 1921, when Sula is eleven years old, her grandmother Eva – ‘operating on a private scheme of preference and prejudice’ – takes in three boys and disregards their given names:
They came with woollen caps and names given to them by their mothers, or grandmothers, or somebody’s best friend. Eva snatched the caps off their heads and ignored their names. She looked at the first child closely, his wrists, the shape of his head and the temperament that showed in his eyes and said, ‘Well. Look at Dewey. My my mymymy.’ When later that same year she sent for a child who kept falling down off the porch across the street, she said the same thing. Somebody said, ‘But, Miss Eva, you calls the other one Dewey.’
When the third one was brought and Eva said ‘Dewey’ again, everybody thought she had simply run out of names or that her faculties had finally softened.
‘How is anybody going to tell them apart?’ Hannah asked her.
‘What you need to tell them apart for? They’s all deweys.’
It’s as if Dewey had gone beyond the conventional function of a name (if it ever really had it, here) and become the word for a certain category of people. The first Dewey is a dewey, the second is ‘another one’, and by the third even Morrison is lowercasing them on Eva’s behalf.