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the toy generates a signal of a much lower power (higher impedance) than a guitar pickup is expected to, that's why it's very quiet. as for it being "spotty", the most likely reason is that the sound you hear from the toy is largely due to the way the tiny speaker reacts to the incoming "lo-fi" signal. if you take the speaker out of the chain, you'll most likely lose the sound you're presumably trying to capture.
tl;dr - you can find a way to preamplify the signal before it gets to the amp, but it will still be "spotty". better yet, install a mic/pickup next to the toy's speaker.
the poster above is misinformed. passive guitar pickups don't draw current from batteries, they generate it when metal strings vibrate over the magnetic pole pieces of the pickup. it's safe to assume the toy does not have pickups and therefore needs to generate the signal itself by drawing current from the batteries. the toy is only expected to output its sound into a tiny, low-power speaker, so an appropriately low-powered signal is generated, which is too quiet for a guitar amp because of a much higher impedance, and sounds "spotty" because the signal is too "lo-fi" to sound good on a larger speaker.
as for what was said about guitars:
>Real electric guitars don't have any batteries
a lot of electric guitars/basses use 9V batteries, but plenty don't. an active pickup still generates current, but that signal is too low-power to go straight to the amp and needs to first go through the built-in active preamp. active pickups generate a low-power signal which is supposedly easier for the active EQ to handle and makes the signal less noisy (debatable). active pickups can be designed with a particular tonality in mind even if it sacrifices the output - it works because the signal still goes through the preamp on the way out. on a fully passive instrument the pickup itself is responsible for generating enough current to produce a signal of an acceptable level, which often results in a "nice tone vs high output" tradeoff situation. often times the pickup itself is passive, but the battery is needed for the EQ. active EQ can give you independent control of low/mid/high frequencies, whereas passive EQ is generally limited to a treble roll-off. some guy made optical pickups, which uses light emiters and photodetectors. those obviously require batteries.
>the pickup would be taking the signals from each string individually
not strictly wrong, but not exactly true. in most pickups there are separate magnet pole pieces that pick up the vibrations of each string individually, but the signal that the pickup outputs is generally one and the same. there are exceptions, i think it was Roland who made a pickup that output the sound of
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