With one huge difference: Trump's America First message was 180 degrees away from Kennedy's Cold War embrace of global leadership.
The combination of homage to Kennedy and subversion of his liberal internationalist vision tells you a lot about what Trump's presidency is going to look like - much more than the populist rhetoric about giving America back to the people.
The key Kennedy allusion came in Trump's issuance of a "new decree to be heard in every city, in every foreign capital, and in every hall of power.
From this day forward, a new vision will govern our land. From this day forward, it's going to be only America first."
The setup of the new decree heard far and wide echoes Kennedy's lines: "Let the word go forth from this time and place, to friend and foe alike ... "
Kennedy was introducing an argument about generational change -- and a corresponding change in US foreign policy. In his speech, the word going forth was that the generation that had fought World War II would now fight Soviet hegemony under the banner of global freedom.
Kennedy thought global freedom served US interests. But that was neither his only goal nor his preferred message.
To the contrary, Kennedy used his inaugural address to tell the world that the US would help its friends to achieve freedom for themselves because it was an inherently valuable goal.
Trump's picture is radically different. In place of a "word" he offered a "decree". That term, with its slightly autocratic air, is telling - almost as if the author wanted to choose language stronger than Kennedy's neutral "word".
Trump's decree flatly tells the world that the Cold War ideals of universal freedom is dead, assuming it was ever more than hypothetical in the first place.
This raises a fascinating question: Why would Trump actively want to tell US allies that he has no interest in them except in so far as they might be useful?
Even if national self-interest is an inevitable feature of international politics, it's usually thought that nothing is to be gained by making self-interest implicit.
One answer is that Trump isn't really speaking to a foreign audience, only a domestic one. If this is true, he was borrowing Kennedy's rhetoric of addressing the world merely to make his speech sound similar to the only very-well-known inaugural address of the modern era.
Kennedy's speech is such a standard piece of American oratory that when you train the speech recognition software Dragon NaturallySpeaking, it gives you the option of reciting the address.
If you were writing your first inaugural address, whether you were Trump or his adviser Steve Bannon or some other speechwriter, you might very well be writing with Kennedy's speech open on your desktop in another window.
The other possibility is that Trump knows the rest of the world is listening, maybe even more closely than Americans are. In this scenario, Trump intends to communicate a strikingly new foreign policy vision.
That vision is isolationism of a kind that has not been taken seriously in US foreign policy circles since before World War II.
It rejects the basic idea to which Kennedy gave voice, namely that the US is the leader of the free world and will bear the burdens and responsibilities of leadership in exchange for its benefits.
On this view, Trump is arguing that the era of US leadership is over. Leaders necessarily must tell their followers that they are looking out for the followers' interest.
Trump knows this very well. The first part of his speech was devoted to telling the American people that they were in charge, not the Washington elite.
But if the US isn't going to lead abroad, it need not insist that it has a universal interest in freedom. Put another way, there's no simpler route to renouncing leadership than saying you don't care about your followers for their own sakes.
Rhetoric is not action. There's no guarantee that Trump will conduct foreign policy in the way his speech heralds. But the point of inaugural addresses is to set a tone. Trump's tone is clear: He is the anti-Kennedy. The Cold War is over. The US does not need anyone else.
- Feldman is a Bloomberg View columnist.