It is a standard procedure for hotels to offer more rooms than they have available on the websites. They may have a long history showing a number of cancellations and or no shows. And they should be prepared to walk an arriving guest if they need to.

It's simple to think extending a late check-out to be a easy thing to do, but given the staffing needs (not all properties have full-time staff to clean rooms at any hour), physical room assignments (some returning guests request specific rooms), and lastly timing of guest arrivals. All these things need to be considered when managing rooms, especially late check-outs.

So there is a good chance you weren't lied to, but given an answer based upon many variables.