I don't know anything specific about the Rakuten offer or how it works, but this is the way an affiliate partner discount usually works. You'll have to read the Rakuten offer if it's different.
With an affiliate discount, you have to start your return through the special portal at the partner website, then go there each time to work on the return. That's how it remembers your discount, likely through browser cache and cookies. If you sign into regular Online TurboTax, it would cache an undiscounted page and cookie.
See this FAQ, including the section that says what to do if the discount is not applied:
FAQ: How do I get my TurboTax affiliate partner discount?
https://ttlc.intuit.com/community/charges-and-fees/help/how-do-i-get-my-turbotax-affiliate-partner-d...
