I think this is a common misunderstanding; people don't realise Paula is just a sample playback chip. They hear it produce 'chip' music and assume it's being synthesised - it's an easy mistake to make.
It's a sample playback chip for sure, but it has some characteristics that aren't that common:
All the channels are played back at independent rates, and the possible rates are all at even divisions of the base clock, meaning that there won't be any aliasing no matter the playback frequency. This would be possible on any sound card, really, except the clock rate of the Paula dwarves the playback rates of modern sound cards, meaning much less frequency resolution for playback without aliasing (or complex algorithms to .
The second thing to consider is the modulation options on it. It lets channels modulate either the playback rate or the amplitude of eachother.
A third thing is that it has programmable volume control independent of output waveform, meaning that even at the lowest possible volume setting, you'll still have full 8-bit resolution.
Besides these things, except for setting the DMA pointers and loop lengths up, the chip is controlled much like a typical PSG of the time.
It might be a misunderstanding to say that it synthesizes sound (depending on the definition of synthesis; technically, if you go by a generic dictionary definition of the word, it does) but it's safe to say that it's a unique chip with some characteristics that distinguishes it from both earlier and later sample playback devices.