Does Thousand Tailed Watcher's effect affect units the opponent plays that turn as well?

No, Thousand-Tailed Watcher's effect does not affect units played after the ability has resolved.

According to the RiftJudge FAQ (ID 6445, 1603, and 7280), Thousand-Tailed Watcher's ability "snapshots" the game state at the moment it resolves. It only applies the -3 Might reduction to units currently on the board at that time. Units that enter play later in the turn are not affected by the reduction.

However, if a unit is played as a reaction to the Thousand-Tailed Watcher's trigger (or to another reaction on the same chain), that unit will be on the board when the trigger resolves and will be affected by the -3 Might reduction (RiftJudge FAQ ID 2236).