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I've been learning some Christmas carols on the guitar and I noticed that the chord structure they have is very different from what I can usually see in popular/rock music.

I noticed that the melody that is meant to be sung is reflected (along with it's pitch) by the chords following this melody.

That feels like the chords don't function here as an accompaniment to a melody and does not provide it's standard function, which is tension and release, but rather follow the sung word changing to match higher or lower pitch that is currently being sung.

So instead of chords being its own thing and melody its own (in a reasonable, consonant way, of course), the chords seem to be the melody itself. If you didn't sing the song and just played the chords, you could pretty much imagine the right words In your head.

It feels different than a typical chord progression I'd sing a song along with.
Is there a name for these two approaches of composing?

And more importantly, what would we call the function of these chords that follows the melody by it's pitch more than by a sense of building release and tension? Are they for an accompaniment, or sometimes maybe it's just ok to follow the melody with chords pitch like and by the function they stand for in the scale?

The video (it's a tutorial in Polish, it's a Polish Christmas carol):

I'm talking about this exact moment (chords changing rapidly trying to match the singing voice) and pretty much entire song in general. The song starts with V (Gmaj) and I believe the entire song is in C so it's already pretty unusual from a popular music point of view to start a song from V chord.

And that's what I mean by "more popular approach to chords accompanying melody":

I'm not certain if this is what you're after, but you may be interested in the concept of the melodic-harmonic divorce in rock music. In short, it's a theory about this repertoire that states that a melody and its underlying harmonies don't always work in tandem the way they traditionally did (like, for instance, in a Christmas Carol), with dissonant pitches often not resolving the way our ears may expect. (If you're interested, you can read more about this phenomenon in this famous article.)

I don't know the history of the Christmas Carol (see also What exactly is a Christmas carol?), but I wonder if these tunes originally began as unaccompanied melodies. If the harmony was added later, it may help explain why it so closely follows the melody.

If I'm understanding the question correctly, it sounds like you are describing what is called appropriately enough, "Chord Melody" which is a popular method to harmonize a melody line often seen in jazz arrangements, but can be used in almost any style of popular music. I'm guessing you got your hands on some of those kind of arrangements, but I can assure you that Christmas carols can be played and sung using the more common chords and separate melody lines also. That said, it generally takes more skill to play "Chord Melody" so it's not used as often.