Short answer: Because, despite being essentially common-practice in structure, it makes ingenious use of traditional (post-shamisen) Japanese tonality.
Long answer: In Japan, the 3rd day of the third month—which means March 3 in the modern age—is Hinamatsuri, a.k.a. Girl’s Day or Doll’s Day. Like all such observances in Japan, Hinamatsuri is associated with several children’s songs. The best known of these is probably “Ureshii Hinamatsuri,” (“Joyful Hinamatsuri”), composed by Kawamura Kōyō 河村光陽 with lyrics by Satō Hachirō. “UH” is also my favorite of the seasonal kids’ songs, mostly because of its distinctly—but not exclusively—old-timey Japanese feel.
The question is, why does it have that feel? Consider the melody for “UH,” presented below in C minor with fairly standard harmonization (I’m not sure what, if any, Kōyō prescribed):
As you can see, it’s in C minor, starts on the fifth, ends on the tonic, is sensibly harmonized in i, iv, and V, and doesn’t even contain any accidentals. Nothing on the page jumps out as exceptional or strange, and yet it sounds unmistakably different from peers like “Koinobori” or “Oshōgatsu.” Why?
Consider the first phrase (mm. 1–2 above). As written, it’s just doodling on G while the harmony works its way from tonic through subdominant to dominant. But it is also exactly the sort of phrase that would be played on a shamisen, or a post-shamisen koto—what I want to call “Shamisen Practice Tonality” (SPT).
I don’t want to get into too much detail about this right now, but suffice it to say this is basically the in scale, except (and this is a crucial difference) viewed as two notes a fifth apart, each accompanied by one note a tone lower and another note a semitone higher. So the example on the Wikipedia page above would be made of [C, D, E♭] and [G, A, b♭].
In mm. 1–2 of “UH” we have the notes [F, G, A♭] and [C]. In other words, it is noodling around the center of one area, with a brief detour to another, higher area. That higher area might be [B♭, C, D♭], or it might be [C, D, E♭]—at the end of m. 2, we don’t yet have enough information to say.
Fortunately, the answer is immediately provided in mm. 3–4, which is in the area [C, D, E♭] with a brief detour up to [G].
What this means is that the first four measures of “UH” also contain only notes we would expect to find in a piece based on SPT, in areas around [G] and [D]. In other words, the tune is both a common-practice melody in C minor, and an SPT melody in G.
Or, almost. The wave function kind of collapses with that final C, which (for me, at least) is so strongly tonic that the SPT feel completely evaporates. It no longer becomes possible to “feel” C as leading up to the D-as-dominant—it’s just C, tonic. (Shifting to a new temporary tonal center a fifth lower than the original tonic was also a thing in SPT, but that’s another story.)
To make a musical analogy, most people reading this are probably familiar with the song “Sakura” (or “Sakura sakura” if you like). Here it is in G—or, at least, using exactly the same notes in much the same way as other SPT pieces that clearly have G as tonic—to match the “UH” sheet music above:
Now imagine if, instead of ending on the D, it went down to C instead:
See (hear) how that final C immediately flips the tonality and whole feel of the piece to “Japanese-influenced melody ending in C minor”? The third measure of that modified version is basically “UH” in miniature.
So, that’s why “UH” has that haunting old-timey sound yet still resolves as we expect it to: Because its C minor melody is built of notes and phrases from the shamisen-derived music of Edo (in G!), but composer Kawamura Kōyō made sure to collapse the ambiguity when he returned to the tonic.