Richard Butler is a good 'splainer, IMHO…
This series of articles is so on point and insightfull.
And I say this as a chip designer working on image sensors (not conventional ones, though).
The concept of ISO really needs a new clarification and wide spreading (of the correct concept). It’s so often misunderstood and misused. Also among who’s working in the field.
When I first started photography, I was scared shitless of high ISO values, I thought the lower the ISO, the cleaner the image so I stubbornly stuck to values under 400.
As time went on, I started embracing noise (not color noise, mind you) so I began parking my camera on auto ISO 100-1600 and forget about it for anything other than macro work, where I would never use anything over ISO 800, but even then, my results weren’t always clean. This went on for a couple of years and I was happy.
Then, just very recently, I noticed something quite strange, for me anyway. I now park my ISO setting on auto 100-1600, even for macro, and I noticed if I have enough light on my subject, even shots at 1600ISO come out rather clean, with very little noise.
Now I don’t particularly understand why and how this happens, but my takeaway is good light is paramount and ISO is a negligible factor.
Light is definitely important. Paramount.
However, I didn’t get if you mean that you also have newer cameras (or if just the time has passed…), but the CMOS image sensor tech hit quite a major improvement on noise levels around 2015ish. I’m not saying there a clear point in time, but for sure, sensors designed from 2016 on have a good chance of being in average much better, noise-wise, than sensors from before.
(I could personally note this passing from the Canon 650D to the 80D)
I’ve been using the same Sony a6000 ever since. But it does have a very decent sensor, especially in it’s price range and considering it’s now a 6 year old camera.
Well, that’s exactly the case. Most of the noise one sees when the ISO is dialed up is the result of that number letting the photographer select an exposure that delivers less light to the sensor. The sensor is really the culprit; at lower light levels, it has a harder time making a consistent measurement of the number of photons hitting the photosite. So, pixel a will count a number, but the adjacent pixel will count a different number for what is approximately the same light energy.
If the term “poisson” doesn’t scare you, this is a good, brief overview:
https://camera.hamamatsu.com/jp/en/technical_guides/photon_shot_noise/index.html
My camera struggles with any ISO above 800. Even 800 is stretching it, unless I am doing something creative or computational. 
I love fresh 
. Fewer opportunities due to covid-19.
As @ggbutcher says, although for photon shot noise I wouldn’t say the sensor is the culprit. Rather, the culprit is light itself. A totally ideal camera isn’t immune to photon shot noise. (However, processing after the exposure is made may reduce the noise in the image.)
I need to understand that. My investigation to date tells me that, for a given scene, two cameras will struggle with the shadows as a function of their respective effective noise floors. Indeed, I’ve experienced that in my migration in the Nikon space from a D50 to a D7000 to now the Z6. Or I think that’s what’s going on; if not, won’t be the first time I’ve been schooled… 
Photon shot isn’t the only source of noise. Some noise does come from the camera’s sensor and electronics. Tests can be done to establish how much noise comes from which source. For example, see Technical Note: Camera Test Protocol.
As a rough-and-ready guide: photograph a white or gray card entirely out of focus, with a large aperture so there is little depth-of-field. Examine the resulting image for variation between adjacent pixels. The result isn’t photographed detail, because there is no detail in an out-of-focus white card. So it is noise.
We can repeat the OOF white card test at different light levels. This provides a reference of how much noise I get in my photos, with my camera, in my normal photography. In ordinary daylight photos at ISO 100, my Nikon D800 shows some noise, but it is too slight to show in ordinary in-focus photos. Hand-held shots at night, with very little light hitting the sensor, have far more noise. Most (but not all) of this is photon shot noise.
Most visible noise in the image is a factor of light. The more light delivered to the sensor, the better the signal to noise ratio, the less visible the noise. So the key to less perceived noise is longer lenses, wider apertures and slower shutter speeds. ISO just amplifies the signal. The higher the ISO, the greater the amplification. As high ISO’s tend to be used in low light situations, all the noise from the low light situations becomes amplified. Thus it appears ISO is responsible for more noise, when really, it is the other factors. If you captured a lot of light and raised the ISO, noise wouldn’t increase. In fact in some instances, camera noise (pattern noise) decreases as ISO increases, though this is less perceived. ISO and Digital Cameras, ISO Myths www.Clarkvision.com
Now that I’ve read @snibgo’s and @Soupy’s responses, I think my confusion is in the consideration of a single camera versus multiple cameras. With respect to a single camera, I get it, it’s the amount of light that is going to affect the presence of noise presented by that camera.
My journey in improving my photography has been largely one of chasing dynamic range. With my D50, I found myself working each image rather aggressively, pulling up shadows and sometimes backing off when I saw what was in there. When I procured the D7000 on clearance, that angst went way down, as I found more range just doing the linear scale prior to tone curving. Then, a windfall allowed me to procure a Z6, and that thing is just amazing. I can now shoot theatrical subjects with a rather slow 70-300mm f4.5-5.6 at 1600 ISO and get images with pretty clean shadows. On the downside, where I worked hard to ETTR with the D7000, I’m less inclined to chase the fine edge of saturation with the Z6, as I know the shadows will be fine even if I’m a stop down from that. Yes, lazy…
This tells the story pretty well:
https://www.photonstophotos.net/Charts/PDR.htm#Nikon%20D50,Nikon%20D7000,Nikon%20Z%206
So camera-to-camera, I still think there’s significant consideration to be given to dynamic range in pushing the noise floor to one’s desires…
Long story short, the Z6 is going to have great dynamic range because its pixel pitch is 5.94 microns. The bigger the pixel, the more light it can capture, so if you are actually delivering that light to the sensor, then they will achieve higher dynamic range and less perceived noise. Not sure what the D70 was, but the D7000 has 4.79 micron sized pixels which should also allow for very good dynamic range, thought not quite to the Z6’s level.
This is true sensor dynamic range, before any tone curves are added.
@Soupy It’s not all about the size (pun intended), the underlying pixel circuitry design (dual gain introduced in Z6) and semiconductor process technology (2019 vs 2005) matter quite a bit too. And the referred PDR plots as defined by that most excellent web resource already take the pixel size variations into account so different cameras can be compared fairly.
@ggbutcher The PDR alone does not give you the full picture. Take a look also at the corresponding input-referred read noise and the resulting shadow PDR improvement. What those tell you is that dual gain indeed kicks in at ISO 800 for the Z6, but also that there is very little benefit for going beyond ISO 800, so you could claw back a whole stop of highlight headroom and brighten in post without loosing shadow detail. Or, if your scene and subject allow it, you can afford another exposure stop for even cleaner shadows.
For the more technically inclined and curious, Emil Martinec also has an excellent explanation here.
@kmilos That site is an excellent resource. Not sure I totally understand shadow improvement, but if its the same idea to what we do when boosting exposure in raw convertors, then that is not true dynamic range, because it doesn’t mean each pixel can capture extra. Just means you can lighten whatever has been captured. Correct me if shadow improvement means something else. I definitely agree read noise is relevant to dynamic range. If you assume read noise as equal, then bigger pixel cameras have more dynamic range, but of course read noise is not always equal.
It does indeed. You can “capture more” precisely because input-referred read noise drops as you increase the analog sensor gain/ISO. The “more” is only for shadow quality, not talking about the total dynamic range of course. This really is before you touch your data in any raw processor.
In addition to Emil Martinec’s explanation, Bill Claff describes it as well below the chart.
The per-pixel dynamic range depends on the size of the pixel, but the image-scale dynamic range is almost completely independent of pixel size. Rather, it depends on the circuitry design (overall tech level), and mostly the size of the whole sensor.
A small pixel might be noisy, but on a big sensor it becomes a smaller contribution.
Compare the A7s2 and A7r2: massively different pixel sizes, but basically the same dynamic range.