Suppose I have Daring Thief out. My opponent attempts to kill it via Doom Blade or another kill spell. In response I untap Daring Thief with Thassa's Ire, and attempt to gain control of, say, my opponent's 4/4 token. There are no responses, and the exchange happens. I now control a 4/4 token, and my opponent controls Daring Thief.
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Yes, if you exchange control of Daring Thief with another creature while a spell targeting Daring Thief is on the stack, the spell will still be targeting the Daring Thief when the exchange completes. The Thief is still the same object when the exchange happens, it just has a different controller.
You might be thinking of the following rule:
There is only one battlefield, not one per player.