To my understanding, Japanese demographics simply mean that x group will likely be the group to enjoy the series the most. The Shonen demographic is, according to Wikipedia, 12-18, so that fits okay for Death Note, I guess. (I watched it at 22 and don't think I enjoyed it any less than I would have when I fell within the shonen age range myself, but eh.)
So I guess there's two ways that demographics can be used, with some malleability depending on the series in question. One version would be "This series is designed for x age group, and anyone outside of that age group will have trouble enjoying it." Blue's Clues and Dora the Explorer would be like this. Perfectly good learning tools for preschoolers, but they don't have much to offer if you're older than four or five.
The second would be "X age group will probably enjoy this the most, but it still has entertainment value for people outside of that group." Dragon Ball and countless anime/manga in general would fall under this heading. Would Dragon Ball be as amazing and exciting an experience now as it would have been when I was 10 or 11? Nowhere even close. Do I still enjoy watching it? Quite a lot, yeah. The same amount that I probably will at 50 or 70 or 90.
That said, the 'Shonen' label for Death Note still feels a bit arbitrary to me. Unlike Dragon Ball (which I don't love nearly as much as I would have as a kid, no matter how enjoyable I find it in the present day), there's no real difference between how much I'd have gotten out of Death Note as a teenager versus how much I liked it as an adult.
I was actually kind of confused whenever Yuli first mentioned Death Note being a shonen series, because I thought the shonen demographic was 6-13. Some people in that age range might like Death Note, of course (especially towards the upper end of that range), but still not the foremost target demographic no matter how you slice it.
For my part, at 6 I would have found Death Note singularly dry and boring and gotten nothing positive out of the experience at all. At 7, I think I would have appreciated the mood and atmosphere, but still generally found it boring; if someone forced me to sit down with them and watch Death Note with them at 7, it wouldn't have been long before I started getting bored and antsy and wanting to run off and play my NES instead. By the 8-10 range, if someone made me sit down and watch Death Note with them, I think I would have enjoyed it (with the level of appreciation increasing each year, and the amount of exposure time required to warm up to it decreasing); and by 11 or 12 I'd have wholeheartedly enjoyed the series from the get-go.
But yeah, the actual age range of 12 to 18 makes more sense.