Can Windsinger target a unit at another battlefield if it's played from hidden?

No, you cannot.

According to Rule 811.1.d.2 and supported by RiftJudge FAQ #10647, when you play a card from a Hidden state, any targets chosen by its "When you play me" triggered ability must be at the same battlefield where that card was hidden.

Because your Windsinger was hidden at a specific battlefield, its targeting options are restricted to that battlefield. It cannot target a unit at a different battlefield when played from its hidden state.