Since the "editor" is from a different journal, I think that their statement is perfectly reasonable and there are no ethical issues here. Academic courtesy, perhaps, but not ethics.
There is no real reason that another party can't publish corrections to published work. After all, it is sometimes impossible for the original authors, having died, to do so. It is courteous, however, to give them the opportunity to do so when possible.
Yes, a publisher should publish corrections to their published work, again, when possible, and it is preferable, not essential, that the originators of the correction be the original authors. However, it sometimes happens that corrections, being judged insignificant, aren't published.
You might check again with that "editor" you spoke with and see if their real intent was, not so much the "website" of the authors, but the website of the publishers of the authors. Personal websites often have little impact or visibility.
They were probably correct in that one publisher might be hesitant to "step on the toes" of another by publishing corrections or by participating in writing a correction in such a case. I think you are free to write such a correction if you wish, but ideally it should be more than a simple correction, somehow extending the original paper if possible. But, again, it seems to me to not be a question of ethics, but courtesy and you could just as well inform the authors and suggest a correction. You could even suggest your participation in the writing.