Gametes are produced through meiosis, which is a reductional division given that produces 4 haploids (n) cells from 1 diploid (2n) cell.
Meiosis is divided in two phases: meiosis I and meiosis II.
Before the meiosis, the DNA duplicates and each chromosome has two sister chromatids, so the cell has 2 sets of chromosomes (2n) with two sister chromatids in each chromosome (4c).
Meiosis I results in 2 haploids (n) cells where each chromosome has two sister chromatids (2c). During prophase I, the homologous chromosomes undergo a recombination process, so the resulting chromosomes are a combination of the maternal and paternal chromosomes.
Meiosis II separates the sister chromatids and results in a total of 4 haploids (n) cells, each chromosome with a single chromatid (c).
The 4 resulting gametes are different from each other and from the original cell they came from, none of the gametes that a person produces is identical to each other, so, even when two siblings share both parents, they will never be identical.