One one hand it's hilariously insecure to begin with, especially now lots of it gets trucked over the internet. On the other hand, there's a number of companies selling access and associated services for trivial amounts of money.
Factual Answer: Logging of IMSI (sim card ID) and IMEI (phone ID). Potentially also logging pTMSI (temporary IMSI used over the air), MSISDN (phone number).
You should be able to get good idea of the handsets distance from the station based on timing delay.
As it's acting as a full fake site it could potentially be used to deny service, drain the battery, MiTM outgoing SMS & phone calls, possibly use the E911 provision to request GPS location from the handset too...
One one hand it's hilariously insecure to begin with, especially now lots of it gets trucked over the internet. On the other hand, there's a number of companies selling access and associated services for trivial amounts of money.