Two answers - yes, and no

WIld and hatachery steelhead can interbreed, so they have chromosomes that are similar enough that they can interbreed. General, that is the definition of what makes a species, though some closely related species, such as horses and donkeys, can interbreed as well.

Having the same DNA is a more precise question. You and I, both being humans, have the same number of chromosomes, and a similar number of genes. Genes are what cause specific features, such as size, color of hair, build, freckles, and such to appear. You and I undoubtedly have different genes than each other. You and your spouse mix genes to create kids, and those kids have different sets, but related genes, to yours. Every gene that the kid has came from one of his parents.

DNA is the material on the chromosome that holds the gene. Think of DNA as a molecule that is a really complex tinkertoy set, with little parking spaces. The tinkertoys in a given parking space describe a gene.

So, hatchery steelhead have DNA that is of a similar length and construction as wild steelhead. The contents of the DNA is different. The content of the DNA between on hatchery steelhead and another is also likely to be different, as well, simply because almost all individuals of a species differe from each other in some small amount.

Hope that helps.