I've had the same thoughts on those. Maybe it also allows them to gather data on whether/how people interact with a webpage itself, which may be an indicator for its quality.
Hmm I’d like to hear more from Kagi on this. They make a point of not logging searches by their subscribers. If this widget is available to non-subscribers outside of Kagi.com, do the same privacy-first principles apply?
See also: https://kagi.com/smallweb