After posting, I tried un-escaping the periods (besides the one in the first capture group), since the Kagi redirects documentation doesn't escape the periods. Surprisingly, that made wiki names with dashes start redirecting correctly.
Here's the string that works: ^https://([^\.]*).fandom.com/wiki/(.*)|https://breezewiki.com/$1/wiki/$2
The question of why my original version did not work remains, but also, shouldn't we be escaping periods in our redirect rules? In regex, a period indicates "any character". Wouldn't a rule that references "reddit.com" also match against "reddittcom.org"? If Kagi is doing anything special with periods to make it so that periods don't have to be escaped, that should be clearly documented.