The property of light compressing in the direction that a photon
emitter is moving on spreading out on the other side should just
change the rate at which the observer receives photons...
The relevant aspect of light here - especially in the classical context Relativity is formulated in - is not the quantization that talking about "photons" makes prominent, but that it is spatially extended waves in the EM field. This means that light emitted from a radially moving source changes the rate at which wavefronts arrive at the observer, and the number of wavefront peaks that pass the observer in a given time are what frequency is, and hence the observed frequency of the light changes.
As mentioned in comments, the intensity also changes because the rate of photon impacts changes but this is a separate and independent effect, and is hard to use in those terms because in practice, the number of photons in an emission of a macroscopic amount of energy is hard to determine.
The wavelengths of the photons being emitted from a source do not
change as the source moves, and so the observer should see the exact
same color of light.
Because Doppler shift of light is a Relativistic effect, you immediately face the question of, as the source moves compared to what? Everyone thinks they are at rest with respect to themselves, and they also think they are "at rest" with respect to the "aether", the proposed analoguous "medium" that carries light in the way that the atmosphere carries sound.
What your teacher said isn't strictly correct, because there's a fundamental asymmetry that exists between sound and light - an emission of sound will change frequency if the source is moving compared to the air around it, as the wavefronts "bunch up" in front of the emitter. (You can imagine this more easily if the source is moving a significant fraction of the speed of sound.) The source can measure the speed of the air whizzing past it and so infer the "true" frequency of the emitted sound as seen in a frame that is at rest w.r.t. the air. This "true" frequency will also change depending on emission direction - emitting in front of you will have the fronts bunch up, emitting behind you will spread them out. A receiver's observation of this "true" frequency is then dependent on its velocity compared to the air, not to the original emitter.
With light, we have a problem because it turns out the aether doesn't actually exist. Everyone, even people travelling at different velocities, thinks they are "at rest" with what ought to be an aether - specifically, they observe no change in the round-trip time in any direction around them, as discovered by the Michelson-Morley experiment. If the aether did exist, then light should take a different amount of time to travel in line with our aether velocity compared to perpendicular to it, and this does not happen.
And if there's no medium, then there's no "true frequency", and so we can't talk about the emitted wavefronts "bunching up" like we did with sound. Unlike with sound, where both source and receiver agree on their speeds relative to the medium, the "true frequency" and thus how they observe the latter, the story of EM Doppler shift splits in two depending which frame you look at:
- In the source frame, you are at rest, and so no shift occurs in the emission. However, if you watch a receiver moving at relativistic velocity through your emitted wave, their observation will be shifted because their own velocity carries them through the wavefronts faster (or slower) than they would be naturally delivered by the light's own velocity if they were "at rest." (...in your frame)
- In the receiver frame, you are at rest, but the source is not. As a result, the emission still travels at the speed of light, and the source still "shoots" them at the correct rate(*), but they end up squashed together (or stretched apart) within the wave that propagates through space because the source has physically moved a significant distance between each wavefront it transmits. It can never overtake the previous wavefront, because that'd involve travelling faster than light, but it can get very close and subsequently emit a wave that has very small spatial gaps between wavefronts, and thus has a high frequency. This is a similar story to the one about Doppler shifts in air, but the key difference is that different observers will disagree on how much "bunching" happened depending on their velocity relative to the source.
(*) If the source is moving very quickly, it might also be undergoing time dilation, but I'm neglecting that for now.
It just so happens that the magnitude of these effects exactly correspond to each other, and both source and receiver can arrive at correct predictions of what frequency the other observed the emission to have. The upshot is that the source not observing its emission to change frequency as it moves, while true, is not the whole story - how the receiver interacts with the wavefronts "already in space" has to be considered in the source's frame.