Single account, multiple media, no access to e-book

I am using Adobe Digital Editions 4.5 and I cannot access the same e-book I purchased on different media with the same account. I use different media/OS at home, during commute and at work: Mac iOS/OSX, Kindle, WinOS.

I try to use the same account on each, but what works on one, does not work on the other and I get error messages like: "The vendor account you entered is nt associated with the item you are trying to open." or "The document is licensed for a different user account".

Q: How to access the same e-book with the same account on different media.