For the forum history: we have been working with drotar in Typekit Support (firstname.lastname@example.org) and determined that this is a bug in Animate CC 2017. Fonts from Typekit are not published correctly from that version of the software. We've asked the Animate team to look into it further.
The workaround for now is to use Animate CC 2015, if possible. That version of the software adds Typekit correctly and consistently to web projects.
This help page shows how to install previous versions of the software from your Creative Cloud desktop application:
This problem has been fixed in the June 2017 update to Animate CC (version 2017.5/build 188.8.131.52). Fonts from Typekit will work as expected in the new version.