Hawking thought - and could "rigorously" deduce from semiclassical gravity - that the information has to be lost because it can't get out of the black hole interior once it gets there.
To see why, look at this "Penrose causal diagram" that may be derived for a black hole solution. Diagonal lines at 45 degrees are trajectories of light, more vertical lines are time-like (trajectories of massive objects), more horizontal lines are space-like.
If you look e.g. at the yellow surface of the star (its world line), you see that it ultimately penetrates through the green event horizon into the purple black hole interior. Once the object - and the information it carries - is inside, it can no longer escape outside (don't forget: time is going up), into the light green region, because it would have to move along spacelike trajectories.
So the information carried by the star that collapsed to the black hole inevitably ends at the violet horizontal singularity and it may never be seen in a completely different region outside the black hole.
The information loss of course depends on the detailed geometry of the black hole that is not shared by a helium nucleus, so it shouldn't be surprised that helium nuclei and black holes have different properties.
As we know today, the information is allowed to "tunnel" along spacelike trajectories a little bit in quantum gravity (i.e. string/M-theory). This process is weak but this weak "non-local process" allows the information to be preserved.
First, I want to make clear precisely what the Chandrasekhar mass is: the maximum mass of a white dwarf supported purely by electron degeneracy. It depends on a few things (notably, the mean molecular weight of the white dwarf) and neglects other sources of or deviations to pressure, but its canonical value of 1.44 $M_\odot$ is a pretty accurate estimate at the mass above which a white dwarf or degenerate stellar core collapses.
The next possible source of support against gravity is neutron degeneracy. i.e. support from the fact that you can't squeeze neutrons into the same quantum state. The Tolman–Oppenheimer–Volkoff limit is the corresponding mass limit for the maximum mass of an object supported by neutron degeneracy. It is much more difficult to calculate this maximum mass because it requires precise knowledge of the equation of state and, currently, we just don't know exactly how matter behaves under those conditions. Even so, the limit is probably more than 2 $M_\odot$ because such a neutron star has been observed and broadly thought to be less than about 3 $M_\odot$. The upper end of the range is very rough, though.
Black holes, however, have no such mass limit because there is no pressure support. We are quite confident that there are black holes at the centres of distant galaxies with masses that exceed 10$^{10}$ $M_\odot$, and nearby M87 hosts a black hole of a few 10$^9$ $M_\odot$. Any apparent limit on the mass of a black hole is just because it hasn't been fed enough.
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You are a little confused in your stellar evolution model. After the ignition of hydrogen fusion in the core of a star, it will next progress to helium fusion, then to carbon/oxygen fusion via the triple-alpha process (I've skipped a lot of steps and details there, if you want the details you can look at either Hansen & Kawaler's Stellar Interiors text or Dina Prialnik's Introduction to Stellar Structure text). What happens next is mass-dependent (using $M_\odot\simeq2\cdot10^{33}$ g and the mass of the star as $M_\star$):
Thus, not every star produces iron in the core; this only applies to stars with mass $\gtrsim8M_\odot$.
The Chandrasekhar limit arises from comparing the gravitational forces to an $n=3$ polytrope (see this nice tool from Dr Bradley Meyer at Clemson University on polytropes)--polytropes basically mean $P=k\rho^{\gamma}$ where $P$ is the pressure, $k$ some constant, $\rho$ the mass density and $\gamma$ the adiabatic index.
That is, in order to find the limit, you need to use the hydrostatic pressure, $$ 4\pi r^3P=\frac32\frac{GM^2}{r}\tag{1} $$ and insert the pressure of the polytrope of index $n=3$ (requires numerically solving the Lane-Emden equation) and then solving (1) for the mass, $M$. If you've done it correctly, you'll find $M_{ch}=1.44M_\odot$.