In "It is me", "me" is not a subject, and it's not an object. It's actually a so called "attribute", also known as "subject complement" (kindof a misnomer since it's not a subject at all).

In Latin, adjectives agree in case with the noun they modify. By extension, it makes sense to put the adjective "blue" in "The flower is blue" in the same case as "The flower" (it already agrees in gender and number, why not case?). By extension, it also makes sense to do the same thing to "a violet" in "The flower is a violet".

So in Latin, attributes take the same case as the noun they modify - usually the subject. A lot of other European languages also have case agreement like that - Russian, German. Old English used to be that way too.

HOWEVER, now English has pretty much lost case. The only standouts are the possessive 's and pronouns taking a different form when they're subject (I, he, she, we, they) versus everything else (me, him, her, us, them). It has also totally lost adjective agreement. Because of this, agreement of attributes with their subjects (or, in some rare cases like "I made him mad", objects) doesn't even make sense anymore. So essentially, English changed rules.

Old English: "It is I"
"I" is an attribute (for the subject) -> agree in case with "It" -> nominative ("I")

Modern English: "It is me"
"me" is an attribute -> not a subject -> not nominative -> "everything else" case ("me")