Can optical fiber communication be used to transmit analog signals?
LEDs can certainly be run at any light output level, and that level controlled very quickly to modulate it. However, the problem, like with any analogue system, is gain, noise and linearity. Optical systems are non-linear, and have uncertain gain in the extreme.
LEDs are somewhat non-linear in current to light conversion, and are temperature dependent. Photodiodes are also non-linear. The fibre couplings can change coupling efficiency with disconnect and remaking, and with temperature, strain, vibration, probably ageing. Lasers are extremely non-linear, having a threshhold. Systems are often intended to be run over any length of fibre up to some maximum. Signal to noise ratio is not good.
All of these point to running the system digitally for preference.
That's not to stop any hobbyist connecting a LED system up and calibrating the linearity, or using over a small enough swing, to run it in an analogue mode. Note that there are opto-couplers designed to be used for transmitting analogue data. These usually have a second photodiode monitoring the output of the LED, used as feedback to linearise the response of the LED+diode combination. The solid coupling between source and receiver mitigates the several problems noted with the fibre part of the connection.