What the others have said is the correct answer to your question in the text.

However I'd just like to add that in the case of an allogeneic hematopoietic stem cell transplant (with a donor) you actually do "inherit the immune system of the donor" - sorta...

When you receive stem cells, those stem cells originate from another body, which the cells are completely compatible with. Should you later down the road need a new kidney for example, the original donor can donate their kidney without the need of immunosuppresants. Normally you'd need immunosuppresants for as long as the kidney was in your body in order to avoid rejection. However since the immune systems cells and the kidney are compatible - you don't.

I suppose it's exceedingly rare though. Just something I remember from immunology classes that I found extremely interesting.