For me, this is a very awkward question to be asked, as at this point in my studies I'm supposed to be quite expert with elementary mathematical tools like Fourier transforms, but this has always bugged me and I've never found a "true" answer to my question, which probably means a lack of elementary concepts at the very root of my knowledge. I've already looked at questions like Instantaneous frequency vs fourier frequency [closed], however I suspect my problem is a little different.
Sorry in advance for the lenght of the question, but I want to try to be as clear as possible in order to describe my doubt. You may as well want to skip to the numbered part if it's too long. Sorry!
In the following, I'll refer to "fundamental frequency" [I later edited the name, as I previously called it, erroneously, "instantaneous frequency", which is why answers are based on that; thanks to robert bristow-johnson and Olli Niemitalo for clearance in the comments] the frequency of any generic sine wave, bolded below:
$$\sin{(\boldsymbol{2\pi f_0}t)}$$
Basically, I feel like anything that has to do with Fourier (series, transform, components of the series, etc.) is the result of a mathematical transformation that can be very useful to look into a signal in a different way, focusing on characteristics obtained from such transformation, that certainly have effects in nature, and are called "frequencies". But at the same time I think of them as a totally abstract mathematical concept, if we compare them to the concept of frequency that I'm used to since middle school: I have a periodic function, and it completes a cycle in 1 second, therefore its frequency is 1Hz. In fact, I know this is the same for Fourier frequencies, as each frequency is that "elementary" fundamental frequency of every sinusoid that composes the signal, and up to now, everything is fine for me. I sum an infinity of sine waves, every one with a specific frequency, and I obtain the original signal.
My problem arises when I think of a rectangular signal (in time domain): ideally, or mathematically, I can easily think of it being the sum of waves such that, the more I sum them, the more the final result looks like a rectangle, and so I can refer to each frequency of those waves and "how much frequency" it "uses". But since this sounds so abstract, I can't really think of a rectangular function having frequencies. I am literally taking a function whose support is bounded and well defined, and it is not periodic. Yet, its Fourier transform is a scaled $\text{sinc}$ function, hence its frequency components are not zero everywhere, thus making it very different from the concept of fundamental frequency which is the one I've always been used to. Should I want to define a frequency for it, I must do it through a Fourier transform, as I can't possibly think of a frequency for an aperiodic signal ("canonically" speaking - that is, referring to the concept of frequency that anyone learns in middle-high school). Sure, I can extend the signal's support by considering the whole time axis and thus making the concept of a frequency more understandable, but that's still something very abstract and would still make this frequency very exotic. This leads me to think that (fundamental) frequency and "Fourier" frequencies are two different concepts that in general behave differently and have nothing in common. And I would be okay with that.
However, it wouldn't be correct, to me, as
- when we use a filter to filter a signal, sometimes we refer to the signal's (and filter's) bandwidth, thus this has to be Fourier's domain of frequencies, so I associate the frequencies filtered out by the filter with the Fourier frequencies only, having nothing to do with the fundamental frequency of a signal, due to what I concluded before; yet,
- when we filter an electromagnetic signal, we consider its instantaneous frequency as the "band" of frequencies to filter in/out; for example, if it is a radio wave, we make sure to consider a filter that passes high frequencies - and this is surely the concept of frequency I've always been used to, as that frequency can be computed using the wavelength of the real EM wave;
- sound is described as a vibration that propagates as a wave of pressure, and that wave's frequency is the sound's frequency. However, a sound is also usually Fourier-transformed in order to look at those frequencies more easily. This kind of suggests that Fourier frequencies are the wave's (fundamental) frequency.
Now, 1. and 2. suggest that in a filter either the term "frequency" is unspecifically used in the Fourier sense and the other depending on the context, creating confusion (to me), or a Fourier frequency has a direct counterpart in the frequency definition used in time-domain. Since 3. seems to confirm that they are indeed correlated, I am led to think that the latter is the right answer. But this totally conflicts with the conclusion I took at the beginning, so in this case I would be even more confused.
Finally, I thought about periodic sine waves and their Fourier series, thinking that their frequency in time domain actually corresponds to one of the only two Dirac's Deltas' frequency in the Fourier series. This would tell me why Fourier frequencies and instantaneous frequencies match when talking about filter applications, but... That's only for sinusoids, as a periodic square wave with period $T$ has frequency $f=\frac{1}{T}$ and its Fourier transform is actually a sum of deltas that sample a $\text{sinc}$ function every $f$, so the frequencies that it is made of are actually more than one, $f$, invalidating the argument. I can see that the first harmonic would be the one corresponding to the instantaneous frequency in time domain, but it doesn't still sound right, as there are still leftover frequencies that don't account for the time domain's one, so to me it would seem that, given an ideal filter and a radio signal with frequency $f$, if I want to get that signal only, I filter out all frequencies that are not $f$, according to the instantaneous frequency, but yet when I talk about "filtering frequencies out" I actually talk about Fourier frequencies, and if I consider only the first harmonic, I am basically discarding some parts of the signal itself (the parts with frequency $2f$, $3f$, etc., in frequency domain), which would make no sense if Fourier frequencies were the same frequencies that we talk about everyday, like the frequency of a cosine wave.
Given all contradictions, there is clearly something I'm missing or that I misunderstood, so... I ask you, what could it be?
Again, sorry for the long question and for it being so trivial, but I feel like I am unnecessarily tripping around. I hope I was clear enough. Thank you in advance and have a good day!