Why do the summoned Servants all belong to different classes? In the Fate universe, is there a reason why the Servants summoned in each Holy Grail War belong to different classes? 

Using relics and additional incantations during the summoning ritual, Masters have some degree of control over the Class (e.g. Matou Kariya's additional incantations to choose the Berserker Class) and exact identity of the Servant. 

What would happen if two Masters both tried the same Class using an incantation (e.g., Berserker), or even used (part of) the same relic to summon the same Servant? For example:


   What would happen if Waver Velvet had only stolen part of Rider's (Iskandar) cloak, unbeknownst to Kayneth Archibald, and both Masters tried to summon Rider?
I think you answered this yourself:


  Using relics and additional incantations during the summoning ritual,
  Masters have some degree of control over the Class


The keyword here is "some". The Servants are summoned by the Grail, even though the Masters have some sort of control of what they are trying to summon. This wiki states (it has a reference too, but the reference is in Japanese):


  The Three Knight classes are the only Servants who will definitely be
  summoned according to the framework of the Holy Grail War, allowing
  for variation in the rest of the classes. More than one version of the
  same class cannot be summoned, even if all the Masters were to use the
  variation in the summoning chant to get a Berserker-class Servant.


So apparently in this case, the Grail will summon a Servant based on it's compatibility with the concrete Master, much as it has happened with


   Caster's master Ryuunosuke Uryuu