No data in Azure SQL after importing bacpac file
TL;DR: It wasn't finished importing yet.

I found out - it was impatience. As soon as I could access the DB and see the structure I thought the import process was done. But it turned out it was still importing. So good to know that you can access it before it's ready. (In hindsight there was sort of an indicator, while it was still importing it showed 1 additional table, something with Transaction... bla).

However, I could have known as I found out that there would still be an 'Importing...' item in the Notifications panel in the Azure portal during the import. It doesn't show that on any of the blades however.