Of course, they CAN be held there indefinitely, in theory. Our office does NOT hold onto decedent remains, unless they are skeletal, and as yet unidentified. I see not purpose in "holding" onto a deceased person for excessive amounts of time. In our jurisdiction, once they are identified, if the family cannot afford to bury them, or if next of kin cannot be determined, the county pays for a burial, with minimal frills (some call it "pauper's burial", but you won't find that term in any of our policies). In rare cases, where the judicial system thinks they know more than us, they will mandate a body not be interred until all questions are settled, something the doctors know is patently ridiculous.
Although decomposition is slowed by refrigerator temperatures, there are changes that take place in the body. It is more akin to mummification than decomposition after the visceral are separated from the body, and they are exposed to low humidity, low temperatures over time.
Every person, regardless of their station in life, deserves full respect in death. While we have our own style of "morgue humor", it is never at the expense of a deceased person.