Because synthetic diamonds are indistinguishable to the naked eye (IIRC a trained eye with a magnifying glass can spot faint nitrogen impurities which are characteristic of natural diamonds) the thing you're really paying for is the piece of paper, the certificate. You're not really paying for carbon allotrope.
So it's less like gold which is fungible and a more natural form of money.
While yes, you are literally paying for the certificate, what you're really paying for is propagandized emotion.
The diamond cartel did a great job equating "real", mined diamonds with love and romance. For some couples, unfortunately, lab-grown just doesn't have the same feel to it.
More importantly, you're paying for the human rights violations. Hopefully, synthetic diamonds eventually become popular enough to destroy the perverse incentives fueling the traditional diamond trade.
So it's less like gold which is fungible and a more natural form of money.
Diamonds feel more like an NFT...