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

Hiraeth? - a Welsh word that has no direct translation

According to Wikipedia, the closest available English translation is homesickness or nostalgia.

Hiraeth (pronounced [hiraɪ̯θ][1]) is a Welsh word which means 'nostalgia', or, more commonly, 'homesickness'. Many Welsh people claim 'hiraeth' is a word which cannot be translated, meaning more than solely "missing something" or "missing home." To some, it implies the meaning of missing a time, an era, or a person. It is associated with the bittersweet memory of missing something or someone, while being grateful of that/ their existence. Hiraeth bears considerable similarities with the Portuguese concept of saudade (a key theme in Fado music), Galician morriña, Romanian dor, Russian toska (тоска), German Sehnsucht and Ethiopian tizita (ትዝታ).

[1] https://en.wikipedia.org/wiki/Hiraeth

My suggestion would be

Sorrow which covers both a sense of loss of the subject and the consequences thereof. As defined in 1b from MW

1a : deep distress, sadness, or regret especially for the loss of someone or something loved
b : resultant unhappy or unpleasant state //to their great sorrow they could not marry

I agree the dictionary definition is not a good context for the question so given the above.

For example, your only sibling dies. This sorrow comes from realizing you will never be an aunt/uncle.

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.