Does it mean that 3mm per square meter was experienced in a specific
area, or does it mean that the total amount of rain had a volume such
that if it was spread over all of Hong Kong the height would be 3mm.
Or is it referred to in terms of per square meter?
They're really the same idea. Fundamentally it means that it filled a rain gauge (of some type) to 3mm height. But every place that gets that experience equally would see the same amount of rain. So if the storm hit the entirety of Hong Kong as it hit the airport, the rain gauge at that airport would record 3mm, and so would any other rain gauge in the city. And (on flat, exposed ground... if water didn't absorb into the soil or evaporate over time) every square meter would get a depth of 3mm... and the region as a whole would have been covered with 3mm deep of water. (Of course water truly flows downhill and is deflected by buildings and such, so you'd end up with much more water in some spots, less in others).
In other words, 3mm is the height of whatever 3-dimensional prismatic shape you want to draw that was equally affected by the storm, be it a box with a square base of 1 m$^2$, a cylinder 1 km in diameter, a surface with a base of the entire city, etc.
Now of course in reality, not every place does experience the storm the same. And so Hong Kong airport may get 3mm, but Cheung Chau may only see 2mm.
And it's still true that a storm that affects a larger area with 3mm of rainfall does put down more total water quantity in entirety. If the same storm hits all of China, that's indeed a lot of water! But of course it's also a larger area to absorb that water. So regardless of whether it's a Hong-Kong sized downpour or a China sized rain system, if it puts down 3mm, it tends to have similar impacts throughout the region affected.
As others mention, a day is a very common time period for reporting such values. But they can be reported for whatever time periods people wish. Often you'll hear values for a specific storm, which can be as short as a quick downpour, or as long as a multi-day typhoon/hurricane. It's whatever the person wants to report about. Usually they'll give the time period or subject. 3mm doesn't fall in 1 second, but it can fall in a period much shorter than an hour. Or it can be the accumulation from only a steady drizzle that lasts days. That's where the rainfall rates (often given in mm or inches per hour) can be worthwhile!