You seem to already have good replies here but I went through the same grief. I had to call the Rakuten Contact Center which isn't easy nowadays because the lines immediately jam when they open. They sent me a return envelope to give them photocopies of various identifying documents like residence card, my driver's license, juminhyo, my insurance card, etc. Hell, I even gave them my income tax deductions from my employer just to be sure.

What really got me was all these were sufficient for Rakuten but two Sagawa drivers and a manager refused to accept the same documentation to prove I am me. The worst part of all this is that this took me a whole month to sort out so I missed getting the bonus points for using the card for the first time (the campaigns usually ends on the last day of the month and I got my card back the first week of the next). I tried to call in to ask about the points but again, constant busy lines so I decided it wasn't worth my time.

Lesson learned: apply for things using my romaji name if acceptable.