Is it possible to deduce a cause from an effect?

There is a logical fallacy which captures the trouble here, called Affirming the Consequent.

Informally, it is the pattern of arguing anything like: "(1) If it is raining, then the ground will be wet. (2) The ground is wet. (3) Therefore, it is raining." This is invalid because there might be other causes of the ground being wet -- like someone spraying water on it from a hose.

More formally, it is the pattern of arguing:

• Premise 1: If A then B
• Premise 2: B
  
• (Invalid) Conclusion: Therefore A

You hinted at the way through this, though: what must be established in your example isn't just that x causes y, but that only x causes y. Here is a valid logical pattern that would reflect that idea:

• Premise 1: If-and-only-if A then B
• Premise 2: B
• (Valid) Conclusion: A

Whether you can establish that y has exactly one cause depends on y: some things do, some things don't.