How does analog video transmission work in toy drones?
In the specs, it says FM.

If you can trust the specs then yes, it transmits both video and audio using FM modulation.

So is the analog video (NTSC/PAL) signal frequency modulated with a
5.8Ghz carrier using analog electronics and power amplified to an antenna?

Probably. The RTC6705 is one IC known to be used in some FPV transmitter modules. It uses a 5.8 GHz VCO (Voltage Controlled Oscillator) and PLL (Phase Locked Loop). It's not completely analog though, as it uses digital counters in the PLL and is programmed via SPI.
You may be able to get more information about a specific device by searching the FCC Equipment Authorization database  using the product's FCC ID.