The caller ID information is sent as a bust of data between the first and second ring. Is is a FSK MODEM signal and therefor must be decoded by a modem circuit. This can be done with an elaborate analog/digital design or with a modem chip and simply not use its other features. There are several formats used for the caller ID data as well so some processing will needed to parse the message. The data rate is 1200 bits per second and includes such things as headers and check sums. The two frequencies used to represent the binary states are 1200 Hz for the Mark (logic 1) and 2200 Hz for the Space (logic 0). The signal level is -13.5 dBm measured at the central office across a 900 ohm test termination.
There may well have been chips made that specifically incorporate this modem function and a microcontroller the parse the data as well as LCD display drivers making a complete system on a chip. Decoding an output from such a chip, if you can get it, may be very difficult. It is easier to use a modem chip and a serial port from a microcontroller that also controls your display or other interfaces.