FastMail is standards-based, so it does not support labels. This is a good thing, and you should stop depending on Google-specific proprietary features. Even when I was on Gmail, I had a lot of issues with labels because the third party mail clients I needed to use didn't support them. The inbox tabs I ended up replacing in Gmail with rules/filters, that moved my social updates, for instance, to an actual social folder which worked properly on third party clients.

That being said, FastMail is also the leading developer/champion of a new mail standard called JMAP, which supports both labels and folders. I suspect, therefore, if it takes off, they may consider supporting labels themselves.