Is it a sin for a man to become attached to a young girl who grows up to be an 11-year-old young woman, while he has not yet been blessed with children and is considered a father to her and she calls him "uncle"?

There is no accountability for a person regarding their inner feelings unless those feelings lead to something that violates Islamic law, for Allah Almighty says: "And there is no blame upon you for that in which you have erred, but [there is blame for] what your hearts have intended." And the Prophet, peace be upon him, said: "Indeed, Allah has pardoned for my Ummah what crosses their minds, as long as they do not act upon it or speak of it."

However, one must beware of the temptation of women, for the Prophet, peace be upon him, said: "I have not left behind me any temptation more harmful to men than women," and "Beware of the world and beware of women, for the first temptation for the Children of Israel was concerning women." If this girl is a stranger to you (not a mahram (definition)), then deal with her on that basis.