When Vesuvan Shapeshifter copies turn face up replacement effects, why do they work?
Compatible replacement effects can chain together.

Vesuvan Shapeshifter's abilities are indeed replacement effects:


  614.1e Effects that read “As [this permanent] is turned face up . . . ,” are replacement effects.


Replaced events do not happen:


  614.6. If an event is replaced, it never happens. A modified event occurs instead, which may in turn trigger abilities. Note that the modified event may contain instructions that can’t be carried out, in which case the impossible instruction is simply ignored.


The event to be replaced is "Vesuvan Shapeshifter is turned face up", and is replaced with "The Vesuvan Shapeshifter is turned face up as the chosen creature". When that happens, the Shapeshifter has not yet turned face up:


  707.11. If a face-down permanent would have an “As [this permanent] is turned face up . . .” ability after it’s turned face up, that ability is applied while that permanent is being turned face up, not afterward.


If the copied creature has its own replacement effect on being turned face up, it is now in effect as it is being turned face up, and it happens as well.


  616.2. A replacement or prevention effect can become applicable to an event as the result of another replacement or prevention effect that modifies the event.


So the sequence of events is: 


You decide to turn Vesuvan Shapeshifter face up
As you turn it face up, before it's actually turned face up, you choose another creature with a replacement effect that replaces getting turned face up.
Before the copied creature is turned face up, its own replacement effect happens and does something.
The copied creature is actually turned face up with its replacement effect applied, as if it was that creature all along.