Was the output of the C64 SID chip 8 bit sound?
Short Answer: yes
The term 8-bit sound is generally not related to any sample size or speed, but to describe the sound 'qualities' of the 8-bit generation of consoles/computers.
So is it correct to say that the audio generation part was really analog,
Each and every sound generation is in the end analogue. A (single) speaker can only produce one, on-dimensional output over time.
The difference between using an DAC with a fixed feeding rate and mixing specialized circuits (like with the SID) is about the amount of data needed to feed either.
Feeding an 8-bit DAC will need one byte per step, so 44 kByte at 44 kHz. A SID based sound system may need only a single byte for the same duration. That is if only a single symmetric frequency is to be outputted. To produce a more complex output, more data is needed. Notable game sounds can already be generated with less than 100 bytes per second.
Sounds like a compression algorithm, doesn't it? And that's the whole idea here. It's more like playing an instrument(*1) than outputting plain analogue levels. Here as well the output is generated using predefined elements, in effect saving much of the rather slim bandwith 8 bit machines had (*2).
Viewed from system design having a sound chip where only sound elements have to be set and manipulated is much like having video controllers with programmable functions (CTIA) and/or sprites (VIC). Instead of having the CPU directly manipulating the bitmap data to create an output image, the image is composed from components which in turn need way less bandwith to be manipulates.
That design view also gives why it got out of fashion - with CPUs fast enough to manipulate bitmap data just in time for output and fast enough to prepare sound data fast enough for straight DAC output, the need for specialized chips vanished.
Or maybe what I am really asking is:
Which is a different question.
If I have an 8 bit, 44 kHz sample based audio chip, could that chip generate audio which could not be distinguished from a real SID chip?
Quality will be the very same as a SID output recorded at 44 kHz - much like a CD with SID-music.
The issue here is, as shown, not some 44 kHz DAC, but a CPU able (aka fast enough) to compute (emulate) the various elements of a SID in time - if not going for calculating it ahead of time (aka batch) that is.
With modern chips/software it's for most parts straight foreward to build a simple SID-alike system. In fact, the web browser your reading this in is already all you need - at least if it supports WebAudio (*3).
*1 - Which is BTW the idea MIDI is based on. Here sound data is encoded as start stop condition for instruments (simplified). As result a rather narrow 31.25 kBit/s (~8 KiByte/s) can be used to produce a whole orchestras sound.
*2 - For example the available bandwith for data transport of a 6502 running at 1 MHz is at maximum less 100 KiByte/s. That's for a copy loop, with some processing is gets way lower. Similar for other CPUs of the same time.
*3 - WebAudio builds its sound generation pipeline from blocks that can be configured to work similar to the SID pipeline. All needed are a few lines JS to configure and link up sound sources, mixers and filters. Of course, this will be only a first iteration. Really remodelling all non linearities/quirks of teh SID will take a lot more.