Or even just regular cameras. If I take a picture of something unrelated, and the background happens to contain a license plate on a car, or the face of a person walking by, am I at risk for recording a person's identifiable information?
No. The so called "Freedom of panorama" [1] will protect you from such things. As long as the license plate is not the main aspect of the image, you should be fine.
Do self driving cars actually record things? As in, store the video? It seems to me they can just process the incoming videostream without storing it, right?
Yes and they upload to the cloud and a lot of vandals have been caught this way.
Did not know that. That's probably not GDPR compliant, no.