After some research, I believe I figured this out.
On each site, there is a job/task running every 12 hours or so (might be bit different, e.g. 10 hours) which will do the following:
- Pick three tags in random from the top 30 tags or so (again, might be bit different, e.g. top 25 tags or 35 tags), top meaning having most questions.
- Cache those tags server side, and use them as the "example tags" when asking new question.
- There is probably a check in place to prevent same tag from being used twice in a row.
The top tags are easily accessible in the tags page (e.g. on English Language and Usage).
To verify this, I have used the Internet Archive, taking https://superuser.com/questions/ask as example and finding a date with two snapshots:
- March 11, 2016 06:22
- March 11, 2016 22:59
As you can see, different set of three tags each time.