If there are supposedly galaxies, or whole other parts of the universe, that we can't ever see due to them receding away from us faster than the speed of light, then why isn't or hasn't this occurred with the cosmic microwave background?
Let me try this from a building blocks point of view.
Imagine the Universe is basically uniform, that is, it's pretty much the same everywhere at any given time, or, at any specific age. We think this is essentially true.
Next, imagine that the Universe is ridiculously, impossibly large, much larger than we can see, because when we look at things very far away, we also look back in time and we can only see as far as the universe is old. We can't see most of the Universe because most of it is much too far away. Given the observed expansion, we'll never see most of the universe. The light from objects too distant will never reach us. We assume there are countless galaxies out there, similar to the galaxies we can see, but we'll never see them.
Lets skip the very early parts when the observable universe was the size of a baseball, or the size of the solar system, because that happened long before the cosmic background radiation and it's not relevant to your question, and we can begin in the, lets say, around 370,000 years of age, when the Universe was much less spread out than it is today.
At that age, about 370,000 years after the big bang, the Universe was still hot everywhere, kind of like fire thinly spread out everywhere. There were no galaxies or stars as they hadn't coalesced yet. Just spread out clouds of fiery gas still spread out everywhere, but beginning to clump together under gravity. Basically just hydrogen and helium nuclei and free electrons, hot enough to be in a plasma state, not too different than flame, and flame is opaque. It's not fully opaque like a solid object, but flame diffuses light, see more details here.
so, 370,000 years of age, the entire universe was kind of a glowing fog. You couldn't see very far in any direction. You could maybe see an AU or a few hundred AU before the opaque plasma blocked your view if you happened to be around at that time, but basically everywhere you looked, you'd just see the heat-glow of the hot universe.
This all changed around the time when the universe was 379,000 years old, during a time called recombination. The universe expanded enough and became cold enough that most of the free electrons rejoined with the atomic nuclei and the plasma became atoms. As a byproduct of this, the universe went from a glowing fog to transparent for the first time. This didn't happen all at once, it was more of a transition into transparency, but for the purposes of this question we can think of this as happening at 379,000 years.
But, because when we look in any direction, the farther away things are the older they are, so at the edge of this transparency, there's always going to be the cosmic background radiation, that was true shortly after the 379,000 year recombination and it's been true ever since. What has changed, is the expansion of the universe keeps stretching the wavelength of the CMB and it keeps moving further away as we move forward in time.
What a person would actually see, when the Universe was 380,000 years old, is the cosmic background radiation in a sphere around them, about 1000 light years distant and in visible light, because the expansion wouldn't have affected it much over just 1000 years. The CMB is kind of like a sphere of time in addition to a sphere of background radiation. It's the furthest thing we can ever see in the EM spectrum, about 379,000 years younger than the age of the universe.
The thing to keep in mind about this sphere is the universe is much bigger than we can see. The glow was everywhere, but the glow we see at any given time is the age of the universe minus 379,000 years old, it's a kind of observable universe bubble, past which we can't see.
The first stars didn't show up until about 180 million years later and the first galaxies, another 20 million years after that, so it was a long time between recombination and the first stars. If a person was around 180 million years after the big bang, they could observe the first stars forming, but only stars that were close to them. Because looking further out looks back in time, so the first stars anyone could observe would be in their neighborhood, but over time, we saw stars form further and further away, but never as far as the CMB.
But what we can see doesn't change what the largely uniform universe did. Stars and Galaxies began to form everywhere, both inside and outside our visible edge of the CMB. Every star and every galaxy is younger than the CMB, but we can infer, even though we'll never see them, that a great many stars and galaxies formed outside or our visible edge of the universe because the universe is much bigger than we can see.
That's kind of a step by step overview, perhaps more wordy than it needs to be. Let me know if that helps.