From the spec: "When clients and servers share a PSK (either obtained externally or via a previous handshake), TLS 1.3 allows clients to send data on the first flight (“early data”). The client uses the PSK to authenticate the server and to encrypt the early data."

The client initiating the 0RTT provides a pre-shared key, thus revealing to the server that they're not a newcomer. I don't know exactly how many bits of that PSK could be used by the server to identify specific clients. For QUIC I think it's a 15-bit identifier. Browsers will need to clear the PSK (and so remove the 0-RTT) when they clear cookies or in a "private browsing" mode.