“I’m a Jedi like my father before me” is a direct callback to " I want to learn the ways of the Force and become a Jedi like my father". I can’t believe this never registered with me consciously until now.
As well as both Luke and Anakin learn the same lessons in their first two films but make a different choice in the third film of their respective trilogy.
The Phantom Menace and A New Hope:
“Feel, don’t think. Trust your instincts.”
Attack of the Clones and The Empire Strikes Back:
“Patience. Think.”
Revenge of the Sith and Return of the Jedi:
Anakin falls to the temptations and succumbs to his emotions while Luke succeeds in finding a rational way to resolve the emotional conflict facing him. In turn Anakin does the same and they both find balance.
It’s like a dialectic. Thesis -> Antithesis -> Synthesis. The middle film in the trilogy offers a counterpoint to what the first film established, then the third film has the protagonist either succeed or fail to reconcile those two conflicting points. For Luke, his initial Thesis was “I will become a Jedi like my father.” The counterpoint of that is learning what an evil man his father turned into. Then the synthesis comes when he resolves to believe in the good in his father, fulfilling the ideal of what his father could have been, which in turn helps redeem Anakin. And he learns from experience that there’s a time to take action and follow your feelings (A New Hope), but there’s also a time to take a step back and think more carefully (Empire).
And, in a way, the OT retroactively offers a counterpoint/antithesis to the prequels. By giving us a similar series of events with an inverted outcome.