Is it possible to donate fat cells to others?

Donating fat from one person to another would cause rejection, infection and other issues. If the Immune reaction of the host or recipient is reduced to overcome the issue of the rejection, then other problems can surface. For practical proposes it is not possible. However, there are groups looking into growing one's own fat outside in the laboratory and then injecting it back to the patient.

There is research being done with stem cells and adipose stem cells from one individual might in the future be utilized for the same person. If fat is harvested from a different person and transplanted into a different recipient, the patient receiving the fat would have to take large doses of medications that decrease their immune respone to prevent rejection of the new material from another human being.  This can cause serious side effects and increases the prevalence of new and potentially serious skin cancers that can become innumerable.

Yes, fat is a tissue not unlike skin or cartilage or bone. IF the immunogenicity of the tissue can be reduced it would be possible. Therefore bone grafts are essentially dead when transferred from one individual to another. You can't simply transfer blood cells without matching blood types. In rare cases, in which individuals are severely traumatized or immunocompromised or immunologically suppressed, they may be able to accept grafts for short periods of time but ultimately they will require long term suppression which has it's risks, including development of cancer, and is generally not worth it in the case of fat transfer.