I agree with Steve about Zelda 2 being a true sequel (and I'm not just saying that because it's my favorite game in the series, I love it). Zelda 2 still captures and uses elements used in the first Zelda game, but the whole view of playing in towns and dungeons is a whole different genre than the original, but it's pretty similar when you consider the other aspects, such as items, gameplay mechanics, music, etc...
However while I do consider Ys III a sequel to I & II since it's officially made that way, I don't consider it as true of a sequel as Zelda II was to the original, because Ys III lacked fewer elements that Ys Book I & II had, unlike Zelda II with Zelda, though there were still a few similarities. And for what Ys III was missing which Ys Book I & II had is why I don't like Ys III nearly as much as a sequel as I liked Zelda II as a sequel, though I still enjoy Ys III.