For a long time, at least a year, Steam Family sharing has been broken on Linux PCs. Fortunately something has changed and it is suddenly working again (as of January 2017).
I'm a bit disappointed at Valve's support, though. The problem hasn't been commented on at all by Valve, even though it has been reported both in the Steam support forum and on github: https://github.com/ValveSoftware/steam-for-linux/issues/4051
At least the problem could have been acknowledged.