No, Sarkhan's ability does not trigger enters-the-battlefield abilities because he is already on the battlefield. He changes type and gains abilities but he is still the same object.
An enters-the-battlefield trigger is what is called a zone-change trigger. Rule 603.6a describes these triggers:
Enters-the-battlefield abilities trigger when a permanent enters the battlefield. These are written, "When [this object] enters the battlefield, ..." or "Whenever a [type] enters the battlefield, ..." Each time an event puts one or more permanents onto the battlefield, all permanents on the battlefield (including the newcomers) are checked for any enters-the-battlefield triggers that match the event.
Sarkhan's ability changes his characteristics. Those are defined in rule 109.3:
An object's characteristics are name, mana cost, color, color indicator, card type, subtype, supertype, rules text, abilities, power, toughness, loyalty, hand modifier, and life modifier. Objects can have some or all of these characteristics. Any other information about an object isn't a characteristic. For example, characteristics don't include whether a permanent is tapped, a spell's target, an object's owner or controller, what an Aura enchants, and so on.
The highlighted characteristics cover everything that Sarkhan's ability sets. More specifically, the ability creates a continuous effect that modifies those characteristics. Rule 110.3 says
A nontoken permanent's characteristics are the same as those printed on its card, as modified by any continuous effects. See rule 613, "Interaction of Continuous Effects."
So Sarkhan is still the same permanent, but once the ability resolves there is a new continuous effect that is modifying its characteristics.