False. Cash is rarely anonymous, but is most always fungible. And it is very rarely censored (because can't).
Well I hate the word anonymity because it doesn't mean a damn thing other than not having a name attached (and what is meant by "name" here?). Unclear to the point of useless. I prefer to say that the chain is obscured or opaque or private. But none of these words are entirely precise.
Cash doesn't have a transcript of previous transections printed on it. If it did, that would cause issues.
Zerocash fully obscures the block chain. Cryptonote doesn't obscure meta-data correlation.