Why is Leia, the twin of Luke, not a Jedi?

She is not a Jedi because she hasn't received any training.

There are hints of her Force powers in the movies.

In The Empire Strikes Back, she senses Luke's distress and has Lando turn around to retrieve him from underneath Cloud City.

Similarly, at the end of The Return of the Jedi, she can tell that Luke made it off the Death Star before it exploded.

In the books that were released later (such as Timothy Zahn's Thrawn Trilogy), it is revealed that Leia has potential for strong Force powers, just like her brother, and has started Jedi training under Luke's instruction.

Somewhat related to This question over on Sci-Fi SE. The short of it is that Ben Kenobi (and Yoda) were keeping tabs from afar on both Luke and Leia. Luke got the nod for Force training for the sole reason that Leia had been captured, and Ben needed some help getting her out of trouble. Ben, pragmatic as he was, used this serendipitous meeting between himself, Luke, and R2 (carrying a message from Leia) to kick-start the plan he and Yoda had set in motion twenty years earlier.

In addition, think back to Ep IV. Vader had Leia in captivity, and was torturing her personally in her holding cell for some time. He never discovered any latent Force-sensitivity in her. Yet, in the heat of battle, in two separate fighter cockpits separated by several dozen meters of vacuum, Vader could sense in just a few seconds that the Force was strong with the pilot in his crosshairs. That's pretty strong evidence that Luke is in fact the stronger of the two in the ways of the Force. This is borne out in EU sources; Leia trains under Luke in the days of the New Republic, and becomes a Jedi Consular in the New Jedi Order using her Force powers for negotiation and diplomacy. However, she is never a warrior to compare with Luke, or even her own children Jaina, Jacen and Anakin, all three of which become Jedi Knights and one of which falls to the Sith.

Because Leia was a woman... it was made 1977... and George Lucas was born in 1944. He wasn't thinking of the sociopolitical context of the story for the future.

Somewhat of a joke, but somewhat serious; cause Lucas didn't think it all the way through yet.

I always sensed he did Ep IV as a simple movie. Once he saw it was big, then Star Wars went from an action romp to a "deep thinking saga" (TM). Think of all the changes in the re-releases that kind of fit this model. Most famously: Han Shot first. Han goes from a rogue in Ep IV and later has to be ret-conned to a hero, firing in self-defense. I remember hearing that there was some dialog about Luke's dad during the attack on the first Death Star that got removed or muted after the Vader-Luke connection was shown.

Somewhat related, I always wondered why in Ep IV Vader didn't sense her Jedi-ness or daughterness right away.

Leia was a Jedi, but gave up her training, as seen in the The Rise of Skywalker. She was the one referred to by Yoda, but you can't be a Jedi without the training. That is why she didn't use the force in the Original Trilogy.