VarBinary(max) updates very slow on SQL Azure
The on-prem instance was just a local install on my surface. The other is a 50eDTU elastic pool on Basic-tier. After the comment I changed it to a standard tier and the query now takes about 2 seconds instead of more than 15!
So I think that's the answer!

I never thought that a local install of SQL on a surface together with all other software like Visual Studio etc running, would out perform SQL Azure by 25 times! (Even on a basic tier). But now I know :-)