And what the difference between STH and SATOSHI?
SATOSHI is also ERC20 token?

SATOSHI ERC20 token can be sent completely anonymously (Zero-knowledge proof). Neither the sender nor the recipient will know each other's addresses.

Is it really possible without changing ETH core? What is the algo or idea of this?

Soon you can try it yourself. A smart contract will be deployed in a few days.

I'm skeptical about this, not sure this is actually possible. But interested nonetheless.

I'll keep the intrigue. Soon you will be able to receive and send tokens on the conditions described above and then your mistrust will go away.