The Milky Way’s black hole isn’t just referred to as “massive” - it’s “supermassive”! But an excellent question nonetheless, as this is a prime example of astronomers using different units interchangeably in a way that can be a bit opaque.
You’re absolutely correct that at the crux of every black hole is an entity called a singularity, which is something of infinite density - a huge amount of mass piled into functionally zero space. If you take the standard method of finding a density, which is “amount of mass, divided by the space it takes up”, this will guide us well for most objects on Earth, but breaks when it comes to singularities. A pound of feathers may weigh the same as a pound of lead, but the density is definitely higher for the pound of lead. Black hole singularities ask us to divide a very large number (its mass) by zero. Dividing by zero will break your calculator, but formally implies an infinite density.
There is a region around the singularity itself which is strongly distorted by the presence of a large amount of mass nearby. Where this distortion is the strongest, space is so warped that in order to escape, you would have to travel faster than the speed of light - an impossible task. Often, this impossible-to-escape region is bundled together with the impossibly dense singularity and referred to broadly as “the black hole”. The boundary of this region - where if you go exactly the speed of light, you go from being not being able to escape, to escaping - is called the Schwartzschild radius. (This is also the boundary known as the event horizon. These two terms are often used interchangeably.)
If you're well beyond this radius, the mass of the black hole mostly behaves like any other mass, regardless of its density, since you’re now far enough away that the physical size of the object doesn’t really matter. However, this radius changes depending on how much mass is packed inside the singularity. The more mass packed in there, the larger the escape-is-impossible meet-your-gravitational-doom region surrounding the singularity is. So to classify black holes, we typically do this by their mass, but mass also controls how big the black hole region is. Classifying by mass also functionally classifies by physical size.
Our broad schema is stellar mass black holes, intermediate mass black holes, and supermassive black holes. This also goes in order from physically smallest to physically largest. Stellar mass black holes tend to be only a few kilometers across- an eight solar mass black hole would be 48 km across, or about 30 miles. That’s driveable, as long as you’re on Earth and not near a black hole. Supermassive black holes, by contrast, are much larger. The one in the core of the Milky Way, if we use its current mass estimate of 4.1 million times more massive than the Sun, is 1.8 au in diameter. (If you placed it where the Sun is, that means it would extend ~90% of the way to the Earth’s orbit. Not...ideal for the Earth.)
So the black hole at the center of the Milky Way, at its very core, is indeed a volumeless, infinitely dense point. But the inescapable region surrounding it is sizeable - measurable on the scale of the solar system.
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