Let X be uniform(0,1) and let Y=X. Clearly the are not independent. Clearly X is uniform. And it's obvious that Y is also uniform.
The comment you quote means "the points are no longer independent of each other, hence each point is no longer uniformly distributed given the others". In you example, Y doesn't have a uniform distribution for X fixed.
Let X be uniform(0,1) and let Y=X. Clearly the are not independent. Clearly X is uniform. And it's obvious that Y is also uniform.
The comment you quote means "the points are no longer independent of each other, hence each point is no longer uniformly distributed given the others". In you example, Y doesn't have a uniform distribution for X fixed.