I believe every province has criteria and a process for people to change their sex designation from male to female or vice versa., but changing to a sex designation other than male or female is yet to be fully addressed.  Although Ontario issues health cards and drivers licenses with the sex designation X to indicate a non-binary gender identity, nobody has issued a birth certificate with the sex designation X yet, and the federal government hasn't announced any plans to allow a third sex on passports either.  

By the way, as to someone's identification matching how they present publicly, I have no idea how a non-binary person is supposed to look and I doubt anybody else does either.