What's the difference between the "me" and the "I" used in the quote below?

But my parents, they lived through the Blitz years, and me, I was sent to a farm.

The sentence above contains the use of an indirect object pronoun as an appositive. Me, you, it, him, her, us, you, them,

Apposition is very common in English, and it is used to emphasize the next word. It is usually used in speech or in recounting a story (narration) to a reader. I am only giving spoken examples, as I am too lazy to look for others or dream them up.

An appositive is a word or group of words that identifies or renames the noun or pronoun that it follows.

Here are some other examples:

"Him, he took the gun and shot her", said John.

"Them, they wandered through my class like sleepwalkers", the professor complained.

"You, you are the one who broke my glasses", the man shouted at me.

"No, it, that stupid dog peed on my living-room rug last night, not the cat", the lady whined.

Please note: with you and it, the pronoun is repeated because the object or subject pronouns are the same thing: you and it.

Their is also a deictic function going on, here but that's not a grammatical feature. It's more stylistics....

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This is an old question, but I disagree with the accepted answer.

This cannot be an appositive because appositives are in the same case as the noun they replace.

So, an appositive would be: "I, The Z, will go home." It would not be: "Me, the Z, will go home."

What is actually going on in this sentence is the omission of some words, possibly "as for" but many things could work here.

But [as for] my parents, they lived through the Blitz years, and [as for] me, I was sent to a farm.

So, there are four clauses here, two independent clauses and two dependent clauses.

They are "I was sent to a farm" and "they lived through the Blitz" because that's how you use those pronouns. When they are the subject, they are in those forms.

It is "[as for] me" rather than "[as for] I" because that is how you use the pronoun after a preposition.

Another reason you know this is not an appositive but a dependent clause is that, in the appositive, either there are no commas or the second noun or phrase is surrounded by commas.

An appositive is a word or group of words that identifies or renames the noun or pronoun that it follows. It is set off by commas unless closely tied to the word that it identifies or renames.

Daily Grammar

In the original text, however, we neither see an absence of commas nor the second noun/phrase being surrounded by commas. We see what we would expect in dependent clauses, a comma separating the dependent clause from the independent one.

Sometimes, the first clause is made into a question as well. For example:

Q: What happened to John? A: [To] him? He was sent to a farm.

Could even be:

A: [You are asking about] him? He was sent to a farm.

Your example from Dutch is probably some form of an appositive because the I stays in the same case. However, English does not seem to have common practice of using an appositive only to repeat the same word (in the same case).