There are Americans that believe the US is one of the oldest countries in the world ie. if you start counting from when nation-states became a thing. Its a super hilarious question from an Indian, Chinese or African perspective. Really almost anywhere in Eurasia.

Same with indigenous people on turtle island (north america).

I'd argue that it doesnt really count since the concept of 'country' though is more colonial in its origins so realistically, a 'country' wouldn't exist until after colonization in these areas.

Nations might've existed with different groups/tribes/villages sharing a culture, but the concepts of clear-cut borders, strict land ownership, and nationalism (which to me are a requirement for the concept of a "country") are significantly different in many indigenous cultures as compared to colonial states. Maybe with the exception of some Latin American empires? Though I don't know enough about those empires to be confident in that.

Isnt there evidence the First Nations came from Indonesia? Or Polynesia?

The Reepublic of San Marino has been independent for a very long time, possibly since 301 ce, that would make it a 1700 years old country that never changed its form of government.

Even more far fetched: Italian fascism considered itself the continuation of the Roman Empire (the middle ages did not count, somehow). Does it mean that they were 2000 years old for a bit? But then, nowadays Italy consider itself the descendant of fascism (with a tinge of shame) but not of the Roman Empire. Can we pick and choose our lineage?

China consider itself ti have been a consistent country since (handwave) basically forever. Is that enough?

Kim Jong Il came in power in 1994.

It's really funny everything from language, main demographic, culture, and dominant religion changed multiple times over just few thousand years.

The only thing that stayed remotely consistent is "ah yes we also have a bunch of cats running around for some reason."

Göbekli Tepe was settled some 10,000+ years ago.

Istanbul was Constantinople in one of the at-least-three other empires that claimed that land. Contiguity is kinda out.

These are actually really interesting examples.

Take Iran:

On the one hand the area where it is located has had organized societies occupying it for an insanely long time.
This would make it one of the oldest countries by this definition.

On the other hand: when asked the question if the current Iran was the same country as back in the 1970's, I would certainly say: "No."
Occupied area and even some of the people might still be the same, but the cultural, societal and governmental characteristics have changed too radically to still be considered the same.
So, by this definition, Iran would be one of the youngest countries.

I think that uncertainty is what OP was getting at:
How much change do we allow to still consider a country the same as at a previous time in history?