Disclaimer: I omit a bunch of hedging about what I'm referring to when I say "classical music" below; think Bach or Brahms
It sounds like you might be coming from a pop/rock background and are familiar with, say, guitar tabs as a way of notating the structure of a song. Other answers have pointed out that:
- in classical music, notation is almost always (from Bach to present day) very precise and labeling chords would be superfluous; the performer just plays the notes on the page (hopefully artfully)
- most music in the classical repertoire can be annotated with (sometimes simple, other times very complex or ambiguous) chord structure. We'd call this harmonic analysis, and it can be a creative process itself (it's something you'd study extensively if you were studying music in college)
- figured bass was a practice for notating semi-improvised parts, and is similar to guitar tabs in spirit
Harmony and harmonic analysis is a really big subject but there are a couple things we can say about popular music, harmony, and notation that might be helpful:
First, the understanding of what a chord really is and what it does is really pretty different between popular music and classical music (think Brahms, say). Some of this is a little tangential to your question:
- in classical music chords are understood to be a sequence of stacked thirds starting from some note of the scale in the key we're inhabiting (or an adjacent key, a secondary dominant chord being a simple example). The way the notes of the chord are laid out across instruments or octaves (how the chord is voiced) is usually a secondary concern and doesn't affect fundamentally how we understand what's happening harmonically in a piece of music. (that's not quite true: we do tend to label the inversion of chords, that is the note that's in the bass. The implication here is that, yes, often the fundamental note of the chord is the lowest note)
- relatedly, classical music is characterized by functional harmony, that is different chords strongly want to proceed on to other certain types of chords (a IV "wants" to go to a V chord, which wants to resolve to the I, etc). This sort of pull underpins the drama and emotional pull of most classical music, and composers rely on the listener to have internalized this sort of musical language
- in contrast popular music tends to be much more loose about the way chords are expected to behave; e.g. both V - IV - I, and IV - V - I are ubiquitous progressions in pop music, but the former simply wouldn't make sense in a classical piece: it would be as if the air just sort of leaked from a balloon.
- Relatedly, pop music tends to use a cyclical chord progression that repeats every 4 bars say; in general this progression can be almost anything. In contrast the structure of chords in a classical piece tends to be longer and more irregular, as the composer leads the listener away from tonic and back again, playing with their expectations based on their familiarity with the language of functional harmony
- Guitarists think of chords in an idiomatic way, that is in a way that is closely tied to the mechanics of the instrument itself: e.g. voicing is of first-class importance, new "chords" arise from needing to allow open strings to ring and it produces a nice effect, etc
Getting back more directly to your question: the way that pop/rock guitarists label chords is in a couple important ways different from the way I would write chords when analyzing a piece by Brahms (say):
Guitarists (and also jazz musicians, where I think a lot of this language comes from) give labels to "chords" which in classical music we don't think of as chords. e.g. "sus" chords like a "C sus4" or whatever; in classical harmonic analysis we don't think of this as a chord; we'd call it a "C major chord" and then talk about a suspension, that is a non-chord tone.
In classical music these suspended notes tend to get resolved; they create tension by being out of place for a moment. The implication is that the listener hears a C major chord, and hears that there's a dissonance. In contrast, in pop music these chords don't need to be passing or resolve, they can just sit there and be an interesting color, effect, or maybe play with the vocal line in an interesting way.
There is a lot of language in classical music for different sorts of non-chord tones, e.g. passing tones, appoggiaturas, etc. If you analyze Bach's 4-part chorales you can see all of these. He'll even use idioms that involve a cascade of suspensions which pass through many chords in complex ways and where at no point can you take a vertical segment and find a pure "unadulterated" chord.
The point I'm trying to emphasize is that a harmonic analysis of a piece of classical music (that is how we'd label it with chords) is often about the tension between what we expect to be hearing as a listener and what notes are being heard, and generally ignore how the chord is voiced or omit some notes that are sounding. Sometimes there are multiple valid interpretations.
I'm sorry this is getting long. Related important concept that should be higher up: counterpoint, the way that multiple melody lines interact to suggest some harmonic structure, i.e. to form chords and do playful things within that chord structure (non-chord tones, above). This is what you see distilled in Bach's chorales say.
Other eras and types of pieces are much less heavy on counterpoint, e.g. a simple Mozart piano piece or a Schubert song accompaniment might have a left hand that strongly outlines the chord structure of the piece, playing arpeggios or the I and V of the chord. It sounds like you're most familiar with that type of writing (and that's what's common in pop music).