Do All Horses Have the Same Color?

Shlomi claims to have proven by induction that in every herd, all horses are the same color:

If there is one horse, then it is the color of itself - thus we have shown that the base case of induction holds.

For the inductive step, we number the horses from `1` to `n`. According to the inductive hypothesis, the horses numbered from `1` to `n-1` are all the same color. Similarly, the horses numbered from `2` to `n` are also all the same color. And because the colors of the horses from `2` to `n-1` are fixed and cannot change depending on how we assigned them to one group or another, then the horses `1` and `n` must also be the same color.

Did Shlomi make a mistake in his proof? If so, find the mistake.