Do you consider being native to a country to mean that your distant ancestors are also from that country?

That is quite literally what native or indigenous means. Becoming indigenous is a multi-generational process when it comes to any species.

If so, how distant is distant enough?

This isn't a fixed number of generations or timescale even if we just limit the scope of this discussion to humans. It has to do with how long it takes for the immigrant population, any existing indigenous populations, and the broader environment to adapt to each other and prosper harmoniously.

I believe that everyone is an individual and should not be intrinsically tied to the actions or birthing location of their ancestors.

This is pure idealism, the idea that we aren't a product of our enviroment. Our family and the culture we are raised in is a powerful force in shaping us as people. Of course we are all individuals, but that is what a population is: a group of connected individuals. And for individuals who break with their family, their culture, or the broader population they are part of, of course over time they will become part of a different family and culture.