In flight mode -- at least in every mobile phone I've come across that has that feature -- all the radios are disabled, so the device is inhibited from transmitting any signal. If you power up an iOS device while already in flight mode, it doesn't even initialise the SIM card until you switch off the mode (if your SIM card has a PIN code set, it won't prompt for the code if powered on in flight mode). So while in flight mode, a mobile phone cannot communicate with any other device, and that includes base stations.
To answer the second part of your question: the phone is traceable by its IMEI, which (in theory at least) is unique. Of course that doesn't mean you know who owns the phone. The SIM card number is also unique, and is a better target for tracing a user as it will be tied to an account with the provider.
 On some devices, you can enable flight mode and then selectively switch on some radios. On iOS, for example, one can use Wi-Fi and Bluetooth in flight mode if these are manually selected 'on'.
 In practice they are almost unique. I have seen devices leave a factory with duplicate IMEI numbers by mistake. It's also possible with some hardware to re-program the IMEI number.
 In some jurisdictions (e.g. the UK), it is possible to buy a SIM card from a shop and use it without the provider knowing your identity.