You pronounce the abbreviated form as “Eye Dee” so you abbreviate it as ID. No, it is not consistent with other instances of abbreviation in English, but half of English isn’t consistent with the other half anyway.

Also, id is an actual word, so that could cause confusion. The earliest uses of the term ‘ID’ or ‘I.D.’ are also from the US military, which absolutely loves abbreviating things and making acronyms for the sake of brevity, even if the shortening doesn’t follow the usual rules.