No, there are DAC chips in certain NES cartridges (the famous SCC-1 from Konami, which appears in games like Gradius).
However Amiga and NES sound chips work differently. The NES chip does not take prerecorded sound, that would be a wavetable synthesis method. the NES uses a PSG (Pulse Sound Generator), it has a couple of waveforms and generates the sounds in real time, thanks to many parameters. The C64's SID chip works teh same, but it's not a PSG chip., it's a bit more complex, a mix of digital synthesizer with analogue resonant filters.
MIDI is just a language for musical machines to communicate
What you hear in PCs is either FM synthesis (old SB cards, the OPL3 chip), and nowadays, they all use wavetable synthesis (of course, these cards also have DACs for samples)
The Amiga uses a DAC which can play digital samples. It does not generate sounds in thsoe tunes you heard, you are listening to really small samples that emulate what a NES or SID chip can do. There were a few trackers that generated its own sounds, SID-chip style (the famous synth sounds, even Octamed included them after a few years). I am not sure how Paula works, so I don't know if these synth sounds generated in real time are being created by Paula or the 68000 chip.
Hope that helps