On an Apple iPhone: Settings > General > About

you will see the Model Name, Model Number, Serial Number, (scroll down), WiFi Phy Address, Bluetooth Phy Address, Modem Firmware, SEID, EID of the phone, then below you will see your SIM card's IMEI, ICCID and MEID number.

So, yes - you are identified VERY SPECIFICALLY, and these identifiers are used on a CONTINUOUS basis not only to track to maintain connectivity between cell towers, but to set SIGNAL to NOISE (SNR) ratio.

This is a ratio such that your signal to the tower, is just strong enough to be recognized over the background noise. Just like talking, you don't sit at a car and scream at each other; you talk just loud enough to be heard. For people, it's common courtesy, for cell phones, it's power conservation.