I couldn't find a datasheet for the ISD1820 so I don't know if it is designed for a dynamic mic, a 2-wires electret or a 3-wires electret mic. All the modules shown in Google have an electret mic on the pcb.
I also do not know which type of mic you used.
I do not know if the speaker amplifier module you bought has an input coupling capacitor to block the DC from the speaker output of the recorder IC module.
Audio impedances are NEVER matched. A low impedance source (the output of the recorder) normally feeds a high impedance amplifier input. Your amplifier module probably needs an input coupling capacitor.
We do not have a schematic for the cheap Ch***** recorder module so we don't know if it has the parts that properly power an electret mic.
Also we do not know the important polarity of the mic pins.
The value of a coupling capacitor is simply calculated from the source and load resistances and the lowest frequency you want. But the resistances are unknown.
Try a film type of 0.33uF to 1uF capacitor from one output pin of the recorder "speaker terminals" to the audio input pin of the amplifier module. Connect the grounds of the modules together. Keep the volume turned down because an LM386 amplifier has low output power.
The module you bought from E-Bay is no-name-brand and does not have a datasheet. Nobody knows anything about it.
The module from Elecfreaks already has an electret mic on it and it is connected properly.
Maybe you can add the parts for the mic that are shown on the Elecfreaks schematic to your module.