The Paula chip simply plays back the sample data at the memory location pointed to by each channel, at the frequency you specify. A sample can be 2 bytes, or as big as available chip RAM, it doesn't matter, but that is all the Amiga can do: play data. The C64 cannot playback samples (without programmer trickery), and can only playback pre-selected waveforms, but has a hardware filter, and other effects such hard sync and ring modulation allowing for very complex tones.
If you playback a short looped sample on the Amiga, there's nothing to stop you changing the data while it is playing, thus allowing complex synthesis, should you have programmed it to do so (see the Sonix music program for a very good example of early Amiga synthesis). The problem with doing this is that it uses CPU time, which in the case of demos/cracktros/etc, could be better spent elsewhere.
So anyway, the Amiga is perfectly capable of playing back music that sounds like a C64, but you need to use the processor to do it. Some people who wrote their own playroutines and didn't use Protracker (Jason Page who wrote Fire & Ice is such a person) used simple methods to synthesize sounds on the fly (the classic C64 PWM lead sound is particularly simple to emulate with very little CPU usage) to improve tone quality, but most people used Protracker which did not have these features.
So chip music in the classic Amiga sense is merely tracker music that uses short looped samples; single cycles of static waveforms that never change in tone. No CPU is utilised to adjust the wave data during playback, so the music is tonally much more simplistic than the C64 (although Protracker does have a simple command to invert sample data while playing - see my Bill's Tomato Game music prehistoric levels for a good example).