When we put a hold on an airline seat for a couple of days, it expires after a couple of days. Now, after couple of days, nobody has booked the seat since I was the one holding it and I did not book it. So why does the agent say that hold has expired? Because by above logic, the seat is still not booked.
When you put a hold on a seat there might be 10 seats available - you held 1, 9 were remaining. You might have put it on hold for $X.
By the time your hold expired - airline might have sold rest 9. So now your once_held_single_seat is now the only one available (on may be a last flight of the day), suddenly this would result in being moved to a higher class for may be $3X.
When your broker says the hold is expired, he might be meaning he can no longer get the same deal for you.
I don't understand what "logic" has to do with it. Placing a hold means that no one else can book the seat for the period of the hold. It is a short-term reservation. So if there are five remaining seats on a flight and you place a hold on one, the airline can only sell or reserve four.
After 24 hours or whatever the hold is, you lose your privilege and the airline can sell all five seats again. That does not mean that it will sell them right away. Maybe no one wants them and the flight is doomed to depart with five seats empty. But your temporary right to that seat is no longer.
Are you asking why the airline would not extend the hold automatically? Well, a hold is very risky for the airline. You have not purchased a fare, and at the same time you have taken away their ability to collect a fare from somebody else. So, they may allow you to place a hold as a courtesy, but it will be temporary, and whether or not they allow you to extend it depends on airline policy and demand.
The term "a hold on a seat" is not very specific, but I'll assume you meant that you put a reservation for a specific flight at a specific fare on hold. After the hold expires, your reservation is cancelled. If the same flight is still available for sale at the same fare, you can put a new reservation on hold and start over. However, if you are using a travel agent, the agency might not want to do it, because the airline may have rules against it and might penalize the agency if they violate the rule.