I would question whether gender identity data needs to be collected at all. If it isn't replacing sex, which it can't, why does anyone need to know? What relevance does it have to anything or anyone else, and how do you know whether it's accurate or not? Unless the purpose of the data is to find out how many people do claim an identity at a particular point in time (eg in the census), so as to see if the number increases or decreases over time, why bother?
In a health care setting, the important information is whether a person is taking cross-sex hormones or has had surgery, and asking for GI doesn't tell you that.
The police and the education system should only concern themselves with sex anyway.