Never heard of that. I got the same impression—that I was wrong in my characterization of "me" as a direct object—from the M-W article.
I'd have to say you got the wrong impression, then. The last paragraph of the entry says,
Clearly, both the it is I and it's me patterns are in reputable use and have been for a considerable time. It is I tends to be used in more formal or more stuffy situations; it's me predominates in real and fictional speech and in a more relaxed writing style. Him, her, us, and them may be less common after the verb to be than me is, but they are far from rare and are equally good.
Here's the short version of the debate:
It is me is the newer form, and it is apparently first attested in the 16th century. Both
it is I and
it is me have existed side by side since then, though in the early 18th century some people started to object to
it is me (namely Bishop Lowth, who also railed against split infinitives and other perfectly standard and grammatical constructions).
Some argue for
it is I on grammatical grounds; because
I is a complement to the subject
it, it should be in the same case (the subject or nominative case). Or so the argument goes. Of course, there's several hundred years of evidence against this argument; English speakers use object pronouns instead of subject pronouns in a lot of different constructions.
In sum, though some misinformed people may give you a bad time about it, it is perfectly okay to use
it is me.
They don't teach you about those things in high school. What are they?
What are subject complements? That's basically just the term for the thing that comes after a linking verb that describes the subject of a sentence, as in
I am Jonathon or
I feel well. A direct object is the complement of the verb rather than the subject.
