In the same way that wifi-enabled devices and wireless APs broadcast their MAC addresses (FF:FF:FF:FF:FF:FF), do cell-phones/smartphones also broadcast such an ID too to connect to a cell tower? That is, besides their MAC address to connect to a wifi.
Your question is little bit vague and general in nature.However,I will explain it in layman terms.
The way your mobile works with a cell tower is totally different from it works or obtain an IP from a DHCP Server.
I will take LTE(4G) for example. The process of connecting to a cell tower/eNodeB is called as attach.You send your information about your mobile and sim card to the eNodeB. In the LTE architecture the component called PDN gateway is the anchoring point to UE/Mobile and it takes care of the IP address allocation to UE.
It is safe to say that you are comparing apple to oranges when you compare Wifi and cell tower.
To answer your question in short,No the cellphone doesn't broadcast its's ID rather it obtain the connection in RACH channel and attempt to attach to the network.