How can “Harold” and “Herald” ever sound the same?

Pretty simple. I would pronounce both words ˈhɛɹəld.

Wiktionary confirms this and even lists the two words as homophones. It has audio so you can hear what it sounds like too.

First, notice how most American speakers also rhyme Harold and herald with Gerald, caroled, imperiled, and double-barreled. This happens because for most speakers of American English, stressed vowels before R lose the tense–lax distinction that exists in words with those vowels that don't have an R following them.

So even though the vowel phonemes in heck and hat and hate all clearly differ from one another, as soon as you go replacing the final consonant with R in each word, those vowels suddenly stop contrasting. Just which final vowel those all collapse into varies by speaker and listener. Some have only a tense vowel there, which yields [ˈheɹəld]; others have only a lax vowel instead, which yields [ˈhɛɹəld].

The same forces are at work in hero as to whether the vowel before the R is the tense one of peek or the lax one of pick. Under tense–lax neutralization, it simply doesn't matter because those two phonemes are no longer distinguished in that position.

See the Wikipedia article on English-language vowel changes before historic /r/ for far more than everything you ever wanted to know about this truly voluminously lengthy area of study.

Second, the vowel in those two words’ second syllables is fully neutralized into a schwa because it isn’t in a stressed syllable. It doesn’t matter whether if stressed it would have been the vowel from called or the vowel from old; it’s always just schwa when unstressed.

Finally, I can’t tell you why your writer claimed haul and hall are pronounced differently, since as far as a I know, both are [hɔl]. The one that’s different from haul is howl, since the latter rhymes with owl.