Breakers open quickly when there is a large overload. A typical breaker can provide more than it's trip current (15 or 20A typically) for a brief period of time.
What you seem to have experienced could well be an overload condition inside the appliance that caused excessive current draw but not enough to open the breaker. The bar/appliance plug connection overheated and failed before the breaker tripped.
You need to check if the appliance is drawing current even if it is turned off.
The other thing is the power bar and plug connection was not good and the extra resistance caused overheating. This is not uncommon and you can often feel heat generated at the plug end of a cord for this reason. However, if the appliance was off there should have been no current draw.
Since the appliance is old you might be better off discarding it.