The last ability triggers only once per turn, even if more than one planeswalker entered the battlefield under your control.

Oath of Liliana is legendary, just like the previous Oaths from Oath of the Gatewatch. If you already control an Oath of Liliana, you can still cast a second Oath of Liliana to take advantage of its enters-the-battlefield ability. It will enter the battlefield, causing its first ability to trigger. You will then choose one to keep, and the other will be put into your graveyard.

Oath of Liliana’s last ability will trigger even if the planeswalker that entered the battlefield under your control is no longer on the battlefield, no longer under your control, or no longer a planeswalker. This is also true if Oath of Liliana wasn’t on the battlefield when the planeswalker entered.