You don't choose a target for Etched Slith's last ability at the time it triggers. Rather, a second "reflexive" ability triggers when you put a +1/+1 counter on Etched Slith this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

If there's more than one type of counter on the targeted permanent or opponent, Etched Slith's controller chooses which counter to remove.

In a case where you put more than one +1/+1 counter on Etched Slith with its last ability (for example, because of the effect of Doubling Season), the reflexive ability will trigger once for each +1/+1 counter you put on it this way.