I don't think Reminders is supposed to work with this. I got excited when I saw my tasks list in the Calendar option in the Alexa app, but it does not seem to sync anything. The news release only mentions calendars, so I don't think reminders is part of this. Would be nice though.So I linked my iCloud account and iCloud calendar integration works fine as far as I can tell.
What doesn't seem to work and I'm not sure if it is supposed to or not, is reminder integration. Alexa can see my reminder lists as I can check/uncheck them in the calendar settings, but if I ask her what's on my calendar for today, she doesn't give me any reminder information.