If I understand correctly, the band gap (between the conduction and valence band) is due to quantization of energy.
Let me start with discrediting a common belief that quantum mechanics is all about quantization. The simplest quantum mechanical system - a free particle, does not exhibit quantization at all. The Hamiltonian in this case is just the kinetic energy $\hat{H}=\frac{\hat{p}^{2}}{2m}$, with a continuous spectrum $E_{\boldsymbol{k}}=\frac{\hbar^{2}k^2}{2m}$ of plane-wave states $\psi_{\boldsymbol{k}}\left(\boldsymbol{r}\right)=\frac{1}{\left(2\pi\right)^{\frac{3}{2}}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}$. The notion of quantization is thus vague in this context and cannot explain bands. Quantization explains other phenomena however, like discrete emission lines.
Now, what I do not understand is, is the band gap the same as the gaps between the different electron energy levels around the nucleus (that is because of quantization of energy too).
So basically, the nuclear shell model describes the gaps between the different electron energy levels where the electron exists as per QM.
And then there is the band gap between the conduction and valence bands.
....
Secondly, the problem of an atom doesn't have much to do with energy bands. Energy bands appear even without talking about atoms or electrons at all. Bands can form solely as a result of a periodic potential perturbing a free quantum system.
You essentially mix two different questions that are otherswise independent
Why bands form.
How electrons behave in such bands.
For (1), bands can form even for systems of bosons. One example for such systems are ensembles of ultracold atoms in optical lattices. In those experiments atoms are cooled down to extremely low temperatures using radiative forces, and then trapped in standing waves produce by couter-propagating light beams. This induces periodic potential on the atoms which results in energy bands. The trapped atoms can be either composite bosons or composite fermions, depending on their total spin, exhibiting completely different effects.
For (2), it just happened in nature that solid objects do produce such periodic potential for the electrons. Because the electrons are fermions and obey the Pauli exclusion principle you get concepts like valence and conduction bands, Fermi level and etc.. All of this, all chemistry and all life - is the consequence of electrons building up when occupying states. But that's just because they are fermions. They don't form the bands. They just live there.
Actually, there is another type of bands forming in solids. Lattice vibrations can be also quantized, creating energy quanta known as phonons. The dispersion of such phonons form similar band-like structures (see the first and third images to the right in this Wikipedia page). However, because phonos are bosons you have a different behavior.
The reason why periodic potential form bands is a different question. The nearly free electron model provides a simple explanation of this (again, the name electron, because electrons in solids were the problem of interest when developing those theories). It applies in cases where the potential is weak compared to the kinetic energy, such that perturbative methods apply. The idea is that a periodic potential contains discrete frequency components. Therefore, it couples states $\psi_{\boldsymbol{k}}$ and $\psi_{\boldsymbol{k}^{\prime}}$ only for discrete values of $\boldsymbol{k}-\boldsymbol{k}^{\prime}$. This breaks the continuous energy spectrum of plane-waves only at certain points, causing gaps to form within the continuum. The following picture illustrates this in one dimension.
