before digital there were analog computers. some are still flying somewhere now.
analog multistage is and can be one or many amplifiers stage means one step in the string of amp to perform a function by gain and or digitaly by diodes gates. in electronics term stages means steps in the evolution of the circuit whatever it maybe.
Recently I developed multistage amplifier while adopting a simple approach.
I took my single stage amplifier and repeat itself to attain the gain that I wanted. Usually the S11 and S22 will remain where they were in single stage while getting 2 x the gain. With some minor tuning in ADS, i got my amplifier design.
Hope this helps.