Seems to me that a unikernel database should be the first application. Databases tend to bypass practically all the facilities of a kernel anyway. It’s often surprised me that they haven’t merged before now.
A modern DB needs efficient interaction with the OS of course, but I'd say based on experience at many companies a bigger challenge than raw efficiency is the ability to implement changes in the db itself. I've worked on 4 or 5 major databases and we can always identify many ways to improve execution, plan selection, various aspects of the db, it's just the giant challenge to alter a big code base that blocks improvements more than OS layers. Improving plan choice can make queries thousands of times faster - but you have to be able to implement it.
Back in the 90s it was common for databases to implement their own file systems or memory management
Maybe not exactly what you're thinking. But somewhat in the same domain.