TLDR: Not as such, but it does give us some constraints on its properties.
A galaxy cluster consists of three major components, from smallest to largest proportion of mass: galaxies (visible with optical telescopes), intracluster gas (visible with X-ray and radio telescopes) and dark matter (not directly observable). Dark matter makes up about 80% of the total mass; of the remaining mass, 90% is the hot gas between the galaxies (not exact numbers, but order of magnitude is sufficient here).
The bullet cluster image:
In the image, we see two accumulations of galaxies centred roughly where the blue colouring is. Overlaid in red is the X-ray emission from hot gas. Note the conical shock front to the right. The blue colour, finally, shows the mass distribution as measured through gravitational lensing.
What does that mean?
We see that gas and galaxies are separated, in contrast to what we see in regular galaxy clusters. The implication is that two clusters have collided. While stars and hence galaxies are essentially collisionless and pass right through each other (there is a lot of space between stars within a galaxy so collisions happen rarely if at all), the same cannot be said for the hot cluster gas. So the gas remained in the centre of the collision while the galaxies passed through.
Now we know that the mass ratio of gas to galaxies is around 10:1. But gravitational lensing (which doesn't care what kind of mass is present) shows us that the majority of the mass in this system is where the galaxies are.
And the dark matter?
We see that the dominant component of the cluster mass is collisionless (it is where the galaxies are). It also does not interact with normal matter except through gravity (else we'd see it).
An exotic elementary particle would fit this description well, but so might primordial black holes (hypothetical low-mass black holes formed shortly after the big bang).
From observing the bullet cluster alone, the dark component might also be made up of quite massive black holes or other massvie dark compact objects -- these can be ruled out by observing microlensing events around our own galaxy.
What about modified gravity?
Mond and similar theories would have to explain why gas of a given mass would affect space-time differently than the same mass clumped together into a star.
Note, however, that there is a known component of dark matter: shortly after the big bang, neutrons that were not bound into helium (or deuterium etc.) decayed into a proton, an electron and an electron neutrino. This cosmic neutrino background is still around, but measurements constraining the upper limit of neutrino rest mass make it highly unlikely that modified gravity theories could use them to explain the bullet cluster. Using our standard theory of gravity, its influence on cluster physics is utterly negligible.
The observation of the bullet cluster provides a big problem for theories that try to explain how our universe works without resorting to some form of collisionless, weakly interacting matter as the major component of the matter in our universe.
It puts some constraints on the nature of dark matter but does not by itself rule out a form of dark matter that is much more massive than an elementary particle.
Galaxy cluster composition
Observing intracluster gas with radio telescopes