In the second quantization approach to quantum field theory, how I understand it, the field is decomposed in components of definite momentum which are treated as non-interacting harmonic oscillators, which are then quantized.
The spectrum of a quantum harmonic oscillator consists essentially of the non-negative integers, and it is natural to interpret the state corresponding to the eigenvalue $n$ to be the state that contains $n$ particles of the corresponding momentum. (Please let me know where you think that my understanding of second quantization is mistaken, if you think it is.)
In other formalisms, especially in the path integral formalism, of course we still have a photon field, fermion fields, etc, but the nicely countable discrete excitations are not explicitly present, for as far as I can tell.
Could we say that the concept of a particle in QFT as a discrete and countable quantity, in other words one that somewhat resembles a particle in our everyday experience, is something specific to second quantization (more generally: the formalism we're working in), rather than a concept inherent in QFT?