The time of dog domestication took place ~15,000 years ago. It was also ~15,000 years ago that the first Native American migrations from Asia took place. These 2 facts places an upper bound on the time of differentiation between Poodle and Chihuahua. I consider 15 to be a "few", hence "a few thousand years".

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https://en.wikipedia.org/wiki/Origin_of_the_domestic_dog

https://en.wikipedia.org/wiki/Native_Americans_in_the_United_States#Settlement_of_the_Americas