You're only considering intrinsic value. That's helpful for raw commodities.
For nearly everything else, value is tied to perception, history, etc.
If I have two identical baseballs, they're both "just" baseballs. But one of them could have much more value due to its history: maybe it was a homerun ball by a famous player, or a ball that I or a family member hit/caught. Same function. Could be otherwise identical, but could also be worth very different amounts.
The stories we tell is where the primary value is.
In this case, your wife values the story of a natural diamond that was formed through long, natural processes and required the effort of finding, excavating, cutting and so on.
Now you could disagree with that story, or you could dislike the tellers and amplifiers of that story, but the fact remains that your wife and many others value that story, making it more precious than merely "shiny carbon."
This is branding and marketing 101. Humans are storytelling machines and they understand value largely via stories and relationships. Misunderstanding that is a failure to understand essential human characteristics.