But what was the second approval for? My main character didn't do anything else. Companions approving their own actions would be kind of cool, actually.
What he's automatically approving here is his father ultimately being saved, which is his task way back from ACT 1.
Some companions have specific tasks and when you do them they'll approve. For example with Lae'zel;
if you do as she says and find Zorru, she will immediately gain approval as soon as you meet him. Same thing happens immediately as soon as you approach the zaith'isk in the creche, since Lae'zel's entire task from the very beginning is to reach a zaith'isk.