Well, one could think of that professor's statement as referring to how that quantum mechanics is needed to describe the basic physics of electrons and atoms in the components that make up the electronic devices used to do signal processing - however the behavior of many components, especially in analogue circuits, at high signal intensities (i.e. currents carrying much higher charge flow rates than a few electrons per second and with energy levels much greater than a single photon of comparable frequency to the signal(s) involved) can be usefully approximated very well with classical mechanics - namely Maxwell's equations. Components like transistors, however, that are often used in digital signal processing do require a more thoroughly quantum understanding to fully make sense of. But I don't think it's necessarily connected to what the video is describing.
Regarding the question of waves versus particles in quantum mechanics and particles versus fields in quantum field theory - all this is explained by the mathematical notion of the Hilbert Space which is used to describe the system(s) in question. All quantum states are members of a Hilbert Space which is associated with the system and it is possible to represent the same space in different (mathematical jargon: isomorphic) ways, in particular, as amplitude functions (wave functions) giving a quantum amplitude for each possibile value of some particular measurable quantity, and which of these you take as representation determines which is "fundamental".
In particular Hilbert Spaces are a special kind of vector space from linear algebra, and in linear algebra every vector space has one or more basis sets which are a "just big enough" subset of the vectors that can be combined together to generate all other vectors in the space - and there is one basis set for each kind of measurable quantity and associated with a certain mathematical operator defined on the space. If you use the position operator, your wave functions will be represented as amplitudes for particle positions, and this corresponds to the particle view. If you use the momentum operator, your wave functions will be represented as giving amplitudes for particle momenta, and this corresponds to the wave view. For quantum fields, if you use the number operator, that is the particle view, and if you use the field operator, that is the field view. Each one is another way of describing the exact same mathematical object by looking at it through a different lense, so they are entirely equivalent and neither one is superior to the other in just the minimum mathematical setup of quantum mechanics.
This is also what is going on in the 3Blue1Brown video - the mathematical space of Fourier-transformable signals and waves is essentially the same (not sure if exactly but one may include the other as a suitably broad subset) as the Hilbert Space used to describe a single particle in 1-dimensional space, and thus both by virtue of having the same mathematical description will have analogous mathematical properties - and it is also this that can be considered as a reason why one would think to talk of the quantum spaces as "particles versus waves". The Fourier transform itself corresponds to a change of basis vectors (isomorphism of equivalent representations) from time to frequency in the signal case and position to momentum in the quantum case. Any time that two such quantities can be considered as different bases of a vector space the uncertainty relation will exist between them - it is no different than the tradeoff of x- vs y-component of a vector in 2D space as you rotate that vector, or perhaps better, hold the vector steady and rotate the 2D coordinate axes about the origin (such a change of axes is effectively a change of basis).