I'm wondering if one can say that a black hole is an object "made of matter" that has a size (as a size, I'm not talking about the size of the event horizon).
I would like to know if one can represent a black hole in the following manner:
If yes, what is the (typical) size of the black hole as an object?
PS: I'm not at all a expert in gravitation theory.
To the best of our understanding, any matter inside a black hole cannot remain at a constant radius. So the sphere of "matter" in your diagram must be collapsing towards the singularity; it cannot remain in the given position.
More properly, what happens inside the event horizon is that particles cannot travel forward in time (which they must do) without getting closer to the singularity. In fact, any particle that enters the event horizon will converge to the singularity after a finite amount of time.
One of the important lessons of quantum mechanics is an operational (as opposed to philosophical) approach to measurement. If there is some property of a system that is really impossible to measure, then assuming that property is well-defined for that system leads to mispredictions about the system's behavior.
While black holes are emphatically not quantum-mechanical objects, the same operational approach applies. A century of research has cemented the idea that it is impossible to observe anything inside of a black hole's event horizon without the observer being trapped forever. Furthermore, an observer within the event horizon can't observe anything interior to themself during their brief, finite-time trip to the singularity. The only experimental way to get information about the size of the singularity is to become part of the singularity; your surviving peers will only see your effect on the event horizon.
In classical general relativity, the size of the singularity is zero. If there is some length parameter between zero and the radius of the event horizon, it comes from physics we have not yet discovered.
Note that a rotating black hole has a ring-shaped singularity. However, the size of this singularity is some fraction of the size of the event horizon, where the fraction depends on the black hole's angular momentum. The event horizon's size remains the only relevant length scale in the problem.
If you were to fall into a black hole then the event horizon is crossed and you would meet the singularity in a finite time on your own watch.
But the region below the event horizon is not necessarily "empty". According to the falling observer, the event horizon marks the point at which light signals can be received from material within the event horizon. That material would have fallen in some time before the observer or subsequently, some time later than the observer.
i.e. Although a falling body and any other material below the event horizon may not maintain a static radial coordinate, they can be observed by other falling observers. So in that sense, if the black hole were accreting material at a high enough rate, then the region below the event horizon would appear to be filled with "stuff" - all of it falling.
However, there is a twist. There would still be darkness in front of you and there is always an apparent horizon between a falling observer and the singularity. What I mean by this, is that there is always a radial coordinate (smaller than the observer's radial coordinate) from below which no light can reach them before they too meet the singularity. Thus you could never see the singularity or see anything else meet the singularity.
Experimentally, we know there exist things that look a lot like black holes. They approach having an event horizon, have a density that matches black holes, and even collide in a way consistent with black holes.
We do not have a way to determine if there is some process that prevents the formation of a singularity and termination of time-like curves on all paths that cross a certain radius.
There are black hole models where no singularity exists; such as ones where the strings of string theory form a surface stretched over the event horizon. Others have a kind of vacuum energy pressure that prevents collapse at some epsilon above critical density to form a true event horizon.
The important thing is that these models are nearly indistinguishable from a classic black hole with a singularity from the outside.
The classic black hole with a singularity has no "inner radius". Everything crossing the event horizon has its world-line end at the singularity in finite proper time; much like there is no way to prevent you from moving into the future, once over the event horizon there is no way to prevent yourself from moving towards the singularity. The singularity ends up being in everything's future.
No persisting structure can exist, because you can't support the stuff behind you, no more than you can build a house that supports something that was there before the house was built.
Now, in the other cases, this doesn't hold. No singularity means there isn't a world-line termination in everyone's future.
https://www.mdpi.com/2218-1997/9/2/88 is an example of such a model.
In such a model, these non-black holes match the observed features of black holes, without having a singularity; something (the vacuum energy) generates a pressure within what would be the event horizon to prevent collapse to a singularity, and a thin membrane near what would be the event horizon corresponds to the event horizon.
This could be described as an alternative to a black hole; instead of a singularity, the end-state of gravitational collapse is something else.
I think the most direct answer is: No, you cannot represent the black hole that way. Anything that falls in goes all the way to the singularity in a very short time (in its frame of reference). It cannot stop and pile up around the singularity.
There are four sections to your diagram:
1 - Dark Blue, the Event Horizon.
2 - Light Blue, undefined.
3 - Yellow, the Black Hole (made of "matter").
4 - Red, the Singularity.
They are all dependent on what is in Section 4, but no singularity exists there. The term refers instead to what we cannot describe with certainty due to observational impossibility. We can't measure these objects directly. Still, it's going to have a definite size because most black holes form when stars collapse. In the final stages of a star's burnout, the elements within will fuse so that they're mostly iron. Without the outward radiation, the iron will collapse under its own gravity and become a dense lump of neutrons--a neutron star.
Angular momentum from this star can keep a black hole from forming if it's a big neutron star with fast rotation but if it slows down, there'll be less centrifugal force to repel photons and they'll be trapped forever. That means an event horizon comes into existence. Within it the dead star continues to spin and a second horizon forms where the lesser centrifugal force allows photons (but not infalling matter) to remain at some distance. Infalling matter gets added to the spinning dead star, increasing its gravity and making the event horizon accordingly larger.
While there's no precisely agreed upon limit to the size of neutron stars, we see can observe a pulsar (radiation emitting neutron star): PSR J1748-2021B at $2.5M_{\odot}$ which is 2.5 times the mass of our Sun. The original star would have been much bigger but matter gets ejected when a supernova accompanies the core collapse and mass is lost. PSR J1748-2021B has a radius of about 18 kilometres. Since it's spinning rapidly, we'd need to slow it down and add a bit more mass to make it a black hole, perhaps one more Sun's worth. This would bring its radius up to about 27 kilometres.
So, there is no actual hole except the region from where light (and matter) cannot escape. That makes for a region that can only be detected by the motion of objects orbiting it, or bright accretion disks if it's in contact with approaching material. The centre is solid and denser nearer the core. If a lot more mass is added, making a supermassive black hole like Sagittarius A* then the event horizon will be much further from the core and your Light Blue undefined section will not be so dense. Remove the Yellow section and all else disappears.
