I used the Pot lid organizer (Israel link / UK link / US link via Amazon) from Ikea with a large plastic hook from 3M.
Luckily, I was in the US a while ago and while strolling the isles of The Container Store (what can I say, some people like to shop for cloth, I like boxes...) I found the best box for washi tapes. The 6-Compartment Unbreakable Box, I bought two of them and I'm happy now (for now...). Look how nicely the washi tape fits in the box.
I tried to find something similar on AMAZON, in case you can't find the exact box, and I think the ArtBin Zerust Anti-Tarnish Box Medium 6 compt. -Clear Art/ Craft Supply Box, 6847AG or the ArtBin 4-Compartment Tarnish Inhibitor Large Box will work as well for washi tape.
Have a great day!