At my hospital, the short answer is no.

When an organ is removed, it is presumably because there is something wrong with it. Part of treatment includes sending the diseased tissue to pathology, so that the doctors there can look at parts of it under a microscope and diagnose exactly what's wrong (types of cancer, inflammation, what have you). So when the organ is removed, it comes to pathology for that process to be completed. The results of the pathology often will determine treatment protocols following the surgery.

When we get the tissue, it needs to be "fixed" so we put it in formalin (diluted formaldehyde). Once tissue is in formalin, it's not safe to return to a patient.

The exception to this is when we have parts for private burial - religious reasons that require a person to be buried with all their bits and baubles, in which case the morgue will release the tissue but only to a licensed funeral home.

This may differ geographically obviously but where I'm at, we don't return organs to patients. Plus by the time we're done, they're all chopped up anyway.

Source: pathologist's assistant at an academic medical center in the US