Let's talk about Riddler's origin and motivation

I don't believe his reasons for being an orphan were mentioned in the story, nor did it play into his motivations.

Basically, he's a sociopath, which you probably could have gathered from his behavior.

His being an orphan did play into what set him off. He was triggered because Thomas Wayne promised the renewal fund to help those in need, which included himself. But after Thomas Wayne died, none of that money went to him and his orphanage. The intended recipients of the renewal fund were basically abandoned after Thomas Wayne's death. At this same time, he was upset that while he and his orphaned brethren (including the babies that froze to death in the winter), the Wayne heir was the media's darling as a poor orphan (a label, which Riddler vehemently challenged because Bruce Wayne was a rich, privileged child).

So the combination of abandonment issues with that jealous, combined with his sociopathetic tendencies led him to try and uncover exactly what happened to that missing money. And he came across the massive conspiracy somehow (I don't believe it's explained how he figured it out). So takes vengeance against the people who put the conspiracy into place, the ones who effectively stole the renewal money. He also lumped Thomas Wayne into all of this because of he was connected to the reporter hit.

I agree that Riddler felt ... off.

I have a nagging suspicion that there is an earlier version of 'The Batman' screenplay where Riddler was actually Hush (a different Batman villain), and he was the orphaned son of the reporter, Edward Elliot, who was murdered by Falcone on the accidental order of Thomas Wayne.

Then it was decided that Riddler had to be the villain (maybe because his name is more recognizable) and a new origin and motivation had to be crowbarred into the script.

He's an orphan because his father was the reporter that was killed before exposing Thomas Wayne.