"You cannot correctly say “This is her speaking,” because speaking is not a noun, and therefore cannot be an object."

"Speaking" is a noun if used as a gerund. It represents the act of speaking, and it is her action. However, even though this may be grammatically correct, I agree that this meaning is slightly different from what the speaker probably intends. I think the intention is that "this" equates to "she/her" (with "speaking", if included, being a present participle describing "she/her"). Therefore, it all comes back to whether an objective noun is appropriate on the trailing side of a linking verb.

I was taught in school that it is acceptable to violate rules of grammar if you do so knowingly to better communicate, not out of ignorance or laziness. If you are speaking in a context where saying "This is she" or "It is I" would incorrectly communicate an air of pretentiousness, then using a less formal form is probably more "correct" in that it better communicates your intention.

Languages are fluid. The rules are always changing. Any choice of words is only as correct as its ability to deliver the desired meaning. It really comes down to knowing your audience.