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I understand that the Dominant 7th chord as a structure is called like that because its chord structure is playable from the Dominant in a Diatonic scale.
However, I noticed that the same structure is playable in the Subdominant of the Melodic Minor scale. In that case it would theoretically be called a Subdominant 7th chord.
So why is the chord's name based on its playability in the Diatonic Scale and not on some other scale?
First some context. Western music began with monophonic Gregorian chant. The most common note sung was the 5th degree, which led to it being named the 'dominant'.
Western music derived it's modal system from the Greeks and others, using a division of the octave into 7 distinct notes using only whole and half steps. Renaissance composers routinely sharped leading tones at cadences and lowered the fourth in the Lydian mode, and effectively only the Aeolian and Ionian modes were in general use by the Baroque, and became known as the natural minor and major scales, respectively.
In the early 1700's, Rameau published "A Treatise on Harmony" in which the modern chord classification was documented, including the building of chords in 3rds and using the highest chord note above the 5th as part of the chord name. This classification was done based on the major and natural minor scales.
Since traditional theory is based on the major scale, and the only degree in that scale that a major chord with a flatted 7th occurs naturally is on the 5th degree (dominant), the chord structure is called a Dominant 7th.
Dominant 7ths can be played on other degrees, and are often called 'secondary dominants', but they have one or more notes in the key altered in order to get the root, major 3rd, perfect 5th, minor 7th structure.
The answer is that the name 'Dominant 7th' is given because the structure of a dominant 7th only naturally occurs on the 5th degree.
The major scale is used as a yardstick in music theory, because there is only ONE form of the major scale. This is convenient, because it leads to consistent labeling: C-A is always a major 6th interval, CEGBb is always a dominant 7th chord , etc. (Edit: I'm talking about the chord structure rather than where it occurs - some will call it a maj/m7 chord if the root is a tone other than the dominant)
If we used the scale of the key instead, we would end up with different labels for the same thing: C-A would be a major 6th interval in C melodic minor, but an augmented 6th in C harmonic minor.
By the way, there are no "8th" chords, because 8 is the same as a root. Chords are always named from the scale of the root. Even if you were using your proposed method of naming in relation to the scale of the key, the 7th note of the chord isn't the 8th note of any scale based on the chord's root.
In a comment you ask
Is it correct that a "dominant chord" (without the 7th) is not necessarily always a Major triad but can also be a Minor triad depending on the scale?
Perhaps, but this question has music history a bit backwards. First there were intervals, then chords, then scales and keys.
There is a common misconception that scales and keys are synonymous. But the three forms of minor scale do not correspond to three forms of minor key. Minor tonality came first, but, long before the concept of keys was developed, there arose a fashion for raising the leading tone in the final cadence. As harmony developed, long after that fashion was firmly established, this gave rise to a rule that the dominant chord in a minor key would have a raised third degree. A few centuries later, that gave rise to the (somewhat artificial) idea that the minor scale has three forms.
While it's true that "dominant" arose as a label denoting the fifth degree of the scale, the term has come to mean the function associated with that scale degree in standard harmonic practice, and that function is the same in both major and minor tonality. If a piece in a minor key has a minor chord on the fifth scale degree, it is likely to be called the "minor dominant," since without that qualifier most would assume a raised third.
The dominant seventh chord got its name from its frequent use as the dominant chord, strongly driving the harmony towards the key's tonic chord. https://en.wikipedia.org/wiki/Dominant_seventh_chord
It's not a law of nature, not physics or mathematics. It's about culture, history, people. https://en.wikipedia.org/wiki/Humanities
There must be many,many more times the dominant seventh harmony is used in major rather than minor keys, over several centuries, so it's far more commonplace. And the same harmony works in minor, using the 5 of the key as root, anyway. True, the same harmony can be found from the melodic minor notes (ascending), so common usage made it the 'go to' dominant seventh chord.
Well, we had to call it SOMETHING! It's logical to go for the most common possibility.
'Dominant 7th' has come to mean a particular chord shape, not necessarily a description of harmonic function in any particular context.
In a simple Blues - using C7, F7 and G7 chords - all those chords might be classed as 'dominant 7th' shapes, though only one of them is the actual dominant 7th chord of the piece.
We might expect that a major 7th chord - C, E, G, B♮ - would get classed as a 'Tonic 7th' shape. But, somehow, that hasn't caught on.
