How does quantization arise in quantum mechanics?

First and second quantization
Quantization is a misleading term, since it implies discreteness (e.g., of the energy levels), which is not always the case. In practice (first) quantization refers to describing particles as waves, which in principle allows for discrete spectra, when boundary conditions are present.

The electromagnetic waves behave in a similar fashion, exhibiting discrete spectra in resonators. Thus, technically, quantization of the electromagnetic field corresponds to second quantization of particles.

Second quantization arises when dealing with many-particle systems, when the focus is not anymore on the wave nature of the states, but on the number of particles in each state. The discreteness (of particles) is inherent in this approach. For the electromagnetic field this corresponds to the first quantization, and the filling particles, whose number is counted, are referred to as photons. Thus, photon is not really a particle, but an elementary excitation of electromagnetic field. Associating a photon with a wave packet is misleading, although it appeals to intuition. (One could argue however that physically observed photons are always wave packets, since to have truly well-defined energy they would have to exist for infinite time, which is not possible.)

This logic of quantization is applied to other wave-like fields, such as wave excitations in crystals: phonons (sound), magnons, etc. One speaks sometimes even about diffusons - quantized excitation of a field described by the duffusion equation.

Uncertainty relation
An alternative way to look at quantization is from the point of view of the Heisenberg uncertainty relation. One switches from classical to quantum theory by demanding that canonically conjugate variables cannot be measured simultaneously (e.g., position and momentum, $x,p$ can be simultaneously measured in classical mechanics, but not in quantum mechanics). Mathematically this means that the corresponding operators do not commute: $$ [\hat{x}, \hat{p}_x]_- = \imath\hbar \Rightarrow \Delta x\Delta p_x \geq \frac{\hbar}{2}.$$ The discreteness of spectra then shows up as discrete eigenvalues of the operators.

This procedure can be applied to anything - particles or fields - as long as we can formulate it in terms of Hamiltonian mechanics and identify effective position and momenta, on which we then impose the non-commutativity. E.g., for electromagnetic field, one demands the non-commutativity of the electric and the magnetic fields at a given point.

Actually in the case of your #2 there is no quantization since the energy spectrum of plane waves is continuous: there is a continuous range of $k$-vectors and thus a continuous range of energies. The wave packet is just a superposition of plane waves, with continuously varying $k$ (or $\omega$) so not quantized.

To highlight the difference I will refer to an old paper of Sir Neville Mott, "On teaching quantum phenomena." Contemporary Physics 5.6 (1964): 401-418:

The student may ask, why is the movement of electrons within the atom quantized, whereas as soon as an electron is knocked out the kinetic energy can have any value, just as the translational energy of a gas molecule can? The answer to this is that quantization applies to any movement of particles within a confined space, or any periodic motion, but not to unconfined motion such as that of an electron moving in free space or deflected by a magnetic field.

I am trying to answer your title question, because your questions about the atomic absorption/emission being quantized, is covered in the other answers.

You are asking "How does quantization arise in quantum mechanics?", and "If photon are wave packets of the EM field, how does one explain the fact that a plane, monochromatic wave pervading all of space, is made up of discrete, localized excitations?".

If you accept that our universe is fundamentally quantum mechanical, then you need to describe the forces that govern it, and you need to describe how the forces act on matter by propagating mediators.

The EM force needs to be quantized to fully describe its interaction with matter. Photons, quanta of light are the only way to describe how light interacts with matter at the level of individual absorptions/emissions.

The weak force is bound by the heavy mediators, the W and Z, and the strong force is bound by confinement, using gluons. Both are in this way fully quantized, when we describe how they act on matter.

In other words, the weak and the strong force are, in some sense, "fully quantum" in that their importance to our world comes completely from their quantized description

Are there weak force waves?

The only exception is gravity, where we do not yet have a full quantum description of how exactly gravity acts on matter by propagating mediators, the hypothetical gravitons. But as you say, the need arises, because we are trying to describe the universe in cases where the gravitational forces are extreme, and dominate over all other forces (singularity).

So the answer to your question is, you can beautifully describe the universe by classical theories, like EM waves and GR waves, if you want to go with big scales, but as soon as you are trying to describe how forces act on matter (exceptions are photon-photon or gluon-gluon interactions) on quantum scale (elementary particles) you need a quantized force.