Learn English – “They never…..I don’t either” or “They never…I do neither.”

There are only a few markers in English which make a question a "negative-polarity" question:

• No
• Not
• Nothing
• Never

And any of their combined/contracted forms ("cannot", "don't", wasn't", etc.)

If and only if any of these appear in the sentence, it is considered a negative-polarity sentence and the "agreeing" response uses "neither":

• I had no idea.
• I was not present at the time.
• I had nothing to do with it.
• I never heard any complaints.

agreeing reply = Me neither!

A prefix on a word which reverses its meaning, even "non-", does not convert the sentence to negative polarity:

He was very non-committal.
- So was she.

You are right that it is not gramatically correct, but certainly in British English it is quite commonly used. It is a double negative, which if we would take literally would mean a positive. But in fact we don't take it literally, and the meaning which it still conveys is

We don't need any education

I would argue that the reason it is used in the Pink Floyd song, is for irony. This is because using no instead of any in a sentence like this, is considered to be something someone uneducated would do. You certainly wouldn't hear the Queen saying it like this!

Also, maybe the single-syllabled "no" just works better for the rhythm than "any".

Not obeying the rules of grammar, in some circles, makes one appear outrageously cool and hip to one's peers. In civilised England however, we recognise that speaking and writing grammatically is more important than anything else in the world.