Free quantum fields are equivalent to a set of independent harmonic oscillators. That's a mathematical fact and nothing is ambiguous here.
Take as the most simple example non-interacting non-relativistic spin-0 bosons. Start with the single-particle case. Its hamiltonian is
$$\hat{H}=\frac{1}{2m} \hat{\vec{p}}^2.$$
There's a complete set of (generalized) momentum eigenstates which are also energy eigenstates.
To make things mathematically unproblematic start first with a regularization by introducing a finite cubic volume ##[0,L]^3## and consider wave functions with periodic boundary conditions, which still allows for self-adjoint momentum operators and thus the infinite-volume limit can be taken in the final results of physical significance.
Then you have a set of momentum-eigenfunctions (I use natural units with ##\hbar=1##),
$$u_{\vec{p}}(\vec{x})=\frac{1}{L^{3/2}} \exp(\mathrm{i} \vec{p} \cdot \vec{x}), \quad \vec{p} \in \frac{2 \pi}{L} \mathbb{Z}^3.$$
The general solution of the Schrödinger equation is given by
$$\psi(t,\vec{x})=\sum_{\vec{p}} a(\vec{p}) u_{\vec{p}}(\vec{x}) \exp(-\mathrm{i} E(\vec{p}) t).$$
Now you want to quantize the field to describe a many-body system of non-interacting bosons.
Via the Lagrangian for the Schrödinger equation
$$\mathcal{L}=\mathrm{i} \psi^* \partial_t \psi + \frac{1}{2m} (\vec{\nabla} \psi^*) \cdot (\vec{\nabla} \psi)$$
you get the canonical field momentum to be ##\psi^*##, and thus the bosonic equal-time field-commutation relations for the field operators (in the Heisenberg picture of time evolution)
$$[\hat{\psi}(t,\vec{x}_1),\hat{\psi}^{\dagger}(t,\vec{x}_2)]=\mathrm{i} \delta^{(3)}(\vec{x}_1-\vec{x}_2), \quad [\hat{\psi}(t,\vec{x}_1),\hat{\psi}^{\dagger}(t,\vec{x}_2)]=0.$$
The decomposition of the field operator in terms of momentum eigenmodes leads to
$$\hat{\psi}(t,\vec{x})=\sum_{\vec{p}} \hat{a}(\vec{p}) u_{\vec{p}}(\vec{x}) \exp(-\mathrm{i} E(\vec{p}) t).$$
The equal-time commutation relations lead to the commutation relations for the annihilation and creation operators
$$[\hat{a}(\vec{p}_1),\hat{a}(\vec{p}_2)]=0, \quad [\hat{a}(\vec{p}_1),\hat{a}^{\dagger}(\vec{p}_2)]=\delta_{\vec{p}_1,\vec{p}_2},$$
where this ##\delta## is a nice Kronecker-##\delta## which makes life easier instead of the Dirac-##\delta## you get for the infinite-volume limit.
This shows that the free QFT is equivalent to an infinite set of independent harmonic oscillators. You can also go on with the analysis of the Lagrangian to get the Hamiltonian for the many-body system (as well as momentum and angular momentum). The Hamiltonian reads (in terms of the annihilation and creation operators)
$$\hat{H}=\sum_{\vec{p}} E(\vec{p}) \hat{N}(\vec{p}), \quad \hat{N}=\hat{a}^{\dagger}(\vec{p}) \hat{a}(\vec{p}).$$
The Hilbert space of the many-particle system is given by the common eigenstates of the number operators ##\hat{N}(\vec{p})##. There's a unique ground state, which is the eigenstate with eigenvalue ##0## for all ##\hat{N}(\vec{p})##, called "the vacuum", ##|\Omega \rangle##. All the harmonic oscillators are in their ground state, i.e., ##\hat{a}(\vec{p}) |\Omega \rangle=0## for all ##\vec{p}##.
The general bosonic number-basis states are given by
$$|\{N(\vec{p}) \}_{\vec{p}} \rangle = \prod_{\vec{p}} \frac{1}{\sqrt{N(\vec{p})!}} \left [\hat{a}^{\dagger}(\vec{p}) \right]^{N(\vec{p})} |\Omega \rangle.$$
All such states are allowed with
$$N=\sum_{\vec{p}} N(\vec{p}) \quad \text{finite.}$$
This is an elegant formulation of the basis of bosonic Fock states, which are cumbersome to write in the "1st-quantization formalism" for a fixed number of particles since you'd have to take care of the symmetrization of the corresponding product states to take the indistinguishability of the bosons correctly into account. This is achieved in the QFT ("2nd quantization") formalism automatically due to the commutation relations of the creation operators.
The very same holds true for fermions (despite the fact that they have in addition a (half-integer) spin). The only difference is to assume canonical equal-time anticommutators leading also to anticommutators for the creation and annihilation operators in the mode decomposition of the quantum field.
