So if no "Hamilton Associates LLC" exists then it can't own the apartment complex, yes?
The key here is what the public tax records show has the owner.
It is possible that when the loan was set up that the LLC was current with its public filing and over time let the filing expire. The loan would never have been approved if the public records were not current. Where LLC's gets into trouble is conducting business without the filings being current under the public notice. It can become an issue for the members should any legal action be filed against the LLC. Its where the LLC can start looking more like a "alter ego" of the members.