When someon is getting a skin graft, does it have to be their skin?

Yes, at least part: Although combinations of grafts (xenografts) have been used, the best results are obtained when at least a portion, especially the keratinocytes come from the patient. The dermis may be obtained from other sources.

Yes: If you use skin from somebody else, the body will gradually reject it (unless it's from your identical twin). That being said, temporary coverage with skin from a skin bank is frequently used - it's just not a permanent answer.