My opponent controls a battlefield with a sprite and a hidden card (sprite call). On my turn I play Windsinger to bounce the sprite. He then reacts with sprite call, and plays a sprite to that same battlefield. This seems fine to me, but my opponent claimed that this counts as a conquer for him. I believe it's not as he already had control of the battlefield. Am I wrong here?

You are correct—your opponent's claim is incorrect. They do not Conquer the battlefield.

According to Rule 464.1, a "Conquer" occurs specifically when a player gains control of a battlefield they did not yet score that turn.

Because your opponent already controlled the battlefield at the start of the interaction, maintaining control (even through a chain of events where units leave and enter) does not constitute a "Conquer."

Why this is the case:

1. Maintaining Control: Under Rule 187.4.c and the associated clarifications, control of a battlefield cannot be lost while there are items pending on the chain. Even if your Windsinger ability bounced their original sprite, their Sprite Call was on the chain. Because the game state did not transition to an "Open state" where they had no units and no items on the chain, they never lost control of the battlefield.
2. No New Control: Since they never lost control, they did not "gain" control. Therefore, the definition of a Conquer (gaining control of a battlefield they did not previously control or score) was not met.
3. Timing Sequence:
   • Chain Link 1: You play Windsinger. Its "When you play me" triggered ability goes on the chain.
   • Chain Link 2: Your opponent plays Sprite Call from hidden as a reaction. Sprite Call (the spell) is now on top of the chain.
   • Resolution: The chain resolves LIFO (Last-In, First-Out).
   • Sprite Call resolves first: They play a Sprite token to the battlefield. They have units at the battlefield.
   • Windsinger's ability resolves next: You bounce their original sprite.
   • Final State: They still have a unit (the new Sprite token) at the battlefield. They held control throughout the entire process and never lost it.

In short, simply adding or replacing units at a battlefield you already control does not trigger a Conquer.