A cool thing to think about is the synthesizer's place in the history of electronic music. There were electronic musical instruments like the theremin well before the synthesizer was invented. Delia Derbyshire famously brought the Doctor Who theme to life without the use of synthesizer. I imagine that she used an oscillator that was perhaps used for calibrating radios, and spliced the individual notes together with pieces of tape, and of course probably drenched the final product in that haunting tape echo >:) And have you ever seen the film The Forbidden Planet or heard its entirely electronic soundtrack? This was before the synthesizer! The husband and wife duo, Louis and Bebe Barron, were creating their own electronic components that produced a tone or altered a sound, and spliced the sounds together with tape.
But the key here is that these were individual electronic components. The synthesizer was the first thing to collect a variety of electronic components together in one place, for the purposes of producing, coloring, shaping, modulating sounds.
Why do we call it an analog synthesizer? Here's how I like to think about it. A guitar produces a sound when a string is plucked, a piano with its hammers, a violin with a bow. In an analog synth, the fast rate of an oscillator's waveform is analogous to the vibrations of a guitar string. Simlarly, the voltage carried by an envelope generator, like an ADSR, can be compared to the hammer striking a piano's strings, or the bow being drawn slowly across a violin.