Each of the Oaths is legendary: if you control more than one of any particular Oath, you choose one to remain on the battlefield and the others are put into your graveyard.

Oath of Chandra’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 Chandra wasn’t on the battlefield when the planeswalker entered.

The “legend rule” checks the full names of legendary permanents. You can control Oath of Gideon and Oath of Chandra at the same time, for example.

Even if you control a particular Oath, you can cast another of that Oath 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, as detailed above, and then its enters-the-battlefield ability will resolve.

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