What is the word for mourning something you never had?
Though a bit broad, the word "melancholia" or "melancholy" can be defined as of a feeling sadness, mournfulness or lamenting with no obvious reason. I think that can include reasons never fulfilled or experienced - deeply rooted unrealized experiences.
https://dictionary.cambridge.org/us/dictionary/english/melancholia
I have been searching for an answer to this very question. I frequently feel this when looking at old photos of people I never met. A sort of grief over the loss of person/place/time period I never experienced and am fully aware that I am probably romanticizing. There seem to be several words which swerve near it, but fail to hit the mark. Hiraeth and Saudade each come close with elements of Weltschmerz and Wabi-sabi (which is not an emotion, but an ascetic; but if the ascetic WERE an emotion, it would fit) thrown in. In researching, I came across this post: https://forum.wordreference.com/threads/nostalgia-for-a-place-or-time-youve-never-been-to.1898243/ in which user EStjarn posits Paranostalia as a good potential fit for a newly coined word to mean this, and I agree.