"Testing Analog ICs" can mean a lot of things.
IC Production Testing which would mean testing each and every part for basic functionality such as power up, and then trimming the parts. This usualy requires specialized equipments and robotic arms to pick and place the IC as this is to be done on each and every part.
IC Characterization. This would be like running comprehensive tests on the IC across different parts from different lots, at different temperatures, voltages, conditions. The data from this would be used to set the specifications in the Datasheet.
IC Debug. This is where one would be trying to pinpoint particular bugs observed in the functioning of the IC. ranging from high current to wrong voltage output etc.
The instruments are mostly the same.
A Scope, A Power Supply, A Voltmeter, An Ammeter, (A Multimeter), A signal generator.
The application would determine what type of each of the equipment you would need. Precision, Speed, Accuracy would determine which one would be best for testing your IC.
The basic equipment would be,
Oscilloscopes: To observe and measure stuff. The more precise and with higher sampling rate the better for precision circuits and fast circuits respectively.
Power Supplies, Multimeters and Sourcemeters. To supply power and measure the voltage and currents. For example for ultra low current measurements of the order of <100nA, one would need a super high precision meters such as 7.5bit precision.
Spectrum Analyzers: For both precision (Such as ADCs) and high speed/RF circuits.
Function Generators or Arbitrary Waveform Generators(AWG) : For driving analog signals. For example to measure the SNR of an ADC one would require a generator with very high linearity to drive the ADC.
Also, if you are going for Temperature testing, one would also need a temperature chamber or other such equipment.