If there are multiple valid interpretations of a piece of music, but
the composer of the music states that only one of those
interpretations is what they intended, are other interpretations still
valid?
This concern is known in literary theory as the intentional fallacy, though some people have discussed it in reference to music as well. The "fallacy" is assuming that there is one true meaning determined only by the intent of the author/composer. However, both literature and music only gain meaning by being understood within a culture/society, and different people in a culture may have different interpretations.
One interesting historical example of this situation is Stravinsky, who continuously (and somewhat notoriously) refined his own biography and his own interpretation of his works throughout his career. If one believes in the "one true composer's intent" theory, what are we to take away from that? If Stravinsky first composed Rite of Spring to correlate with the novel choreography of the first performances, but later Stravinsky claimed he wrote the Rite of Spring as "absolute" music to be understood and analyzed independent of any staging, which Stravinsky are we to believe -- the young composer or the older reflective composer? Are we supposed to analyze Rite of Spring in terms of how it may have worked in trying to portray a staged ballet telling a narrative story, or should we believe older Stravinsky and think of it as a concert piece meant to be understood on its own independent terms? (I mention this specific example because it has been debated by music historians as an example of the difficulty in sorting out composer's "intent.")
So what, then, is a "valid" interpretation? As long as it fits the music and you can convince other people that it makes sense, it might be a good interpretation. A word like "valid" sounds more like a philosophical/logical conclusion, so I don't know how to evaluate whether a particular interpretation qualifies as "valid." That depends on the assumptions underlying your theory of analysis. But, in general, I'd say a successful interpretation should communicate something that makes sense to others, just as a successful piece of music usually communicates something to an audience.
Suppose someone's music has an E♭ chord in the key of C. Person A, the
composer, says they intended modal mixture when writing the chord.
Person B says that they heard it as a tritone substitution. Is person
A more correct than person B, or can music have multiple meanings
depending on context?
Well, first we need to consider whether both analyses could be correct. As other answers have already noted, there doesn't seem to be a tritone in an E♭ chord, so it's unclear how it could be a "tritone substitution." If it's actually an E♭7, and the G-D♭ tritone is substituting for a place where a C♯-G tritone might occur in C major (e.g., substituting for an A7 that is replacing V/ii), then it might be a tritone substitution. But in that case, the function is probably not simple "modal borrowing."
So, perhaps that's not the best example. But there are clearly places where we could debate how to interpret a particular chord. The function/identity of the Tristan Chord is a notable example where various analysts have disagreed about how the chord functions or even which notes are the primary tones of the chord. And yes, the notes of the Tristan Chord could have different functions depending on context -- in another voicing and spelling, it could function as a simple half-diminished seventh resolving in a leading tone function. But that's not how the specific voicing tends to resolve in Wagner's opera.
Anyhow, yes, there are definitely places where different analytical interpretations could make sense. In those cases, the different interpretations will often convey different attributes or focus on different features or aspects of the segment of music in question. (For example, the varying interpretations of the Tristan Chord often depend on a particular analyst's priorities in how to think of harmonic progression/function or even how to choose what is a "chord tone" vs. a non-harmonic one.)
Or another situation, where an old piece of music takes on a new
meaning over time. Does the meaning of the piece then change, or is
there only one true significance?
I already addressed this a bit above with the Stravinsky example. But we could get even more extreme. Consider the application of Roman numerals to the analysis of the music of Bach. Bach never heard of Roman numerals for harmonic analysis. They didn't exist in his time. That's clearly not how he would analyze or would have understood his music. And yet there are hundreds of music theory textbooks that teach Roman numeral analysis often by partly analyzing works of Bach. Are they all "wrong"?
No, I wouldn't say so. They are trying to find patterns that are meaningful in Bach's music, and they are trying to classify those patterns by using an analytical system that was developed later. Most music theory is like this: a lot of analytical concepts were developed in the past couple centuries but they are used to analyze music that predates them. One common example of this is the notion of "sonata form," which was first clearly articulated in the way we talk about it today by Reicha in the 1820s and then really formulated in its modern form by A.B. Marx around 1840. Yet we use it all the time to analyze works by Mozart and Haydn written many decades earlier. In fact, we often then encounter "analytical problems" when we find this piece of Mozart that doesn't seem to "fit" into sonata form, and we ask why it doesn't. One answer (rarely discussed in undergraduate courses) is -- well, Mozart didn't know about sonata form (at least in the terms we discuss it today), so was that even a priority for him?
Heading down this road too far, though, can lead to madness. One begins to question whether any musical analysis of a historical work can be valid without knowing precisely what a composer was thinking, what the understanding of musical concepts of that time were, etc. Those may be valid concerns, but music theory is also about later interpretation and trying to build a model to understand music.
Roman numerals are a model to understand Bach. Sonata form is a model to understand Mozart. They can provide interpretation, but they have limitations. And one has to be very cautious about asserting things like, "Bach was avoiding X harmonic move" unless we actually understand how Bach would have understood harmony. But we can instead make an assertion like, "This moment in Bach seems to go to option Y rather than the more common harmonic move X" and support that assertion with Roman numerals, if we understand what the assumptions of the Roman numeral system are and know where various "harmonic moves" are "common." We may even find patterns Bach wouldn't have even noticed using his theory of harmony, but they become very clear when we apply a novel method to analyzing his music. Maybe those patterns were just something he "intuited" or a function of his training, but something he wouldn't be able to articulate consciously. (See, for example, Michael Polanyi's concept of tacit knowledge.) Yet they may be useful patterns to notice nonetheless.
The bottom line is that we can't help hearing historical music through modern ears. We tend to use the tools and models we know to analyze historical music. That means that interpretations will change over time. Just as musical styles change over time. Just as compositional practices change over time. Just as language changes over time. Music analysis is just a cultural act of interpretation, and it finds meaning through social consensus, just as language does.
Trying to claim that there is only one true meaning/interpretation of a piece of music is like trying to claim there is only one true meaning of a word, and that new-fangled way the young kids are using it today is "wrong." Maybe. But wait 10 or 20 years, and it may be acceptable. Wait a century or two, and your "correct" meaning may be an archaicism no longer understood by most people, while the new-fangled meaning is the meaning.
Artistic works, like any product of human culture, are always evolving with that culture. The same thing is true of their interpretation.
