Presumably when we first sign up at the Adventurer's Guild at the start of the game we're given some kind of registration papers identifying us as an 'official' adventurer of that guild, given that adventurers are often entrusted with important military and political tasks. This would probably extend to when we enlist in a Grand Company we're also given identification for that as well. But it's never actually shown on screen so is more just something that is assumed to happen in the background that game just passes over for simplicity's sake.

There is one point in the story though where we are clearly shown being given identity papers that prove our association with a certain group - upon being granted asylum in Ishgard early in the Heavensward story, the player's character along with Tataru and Alphinaud are handed identity papers with the House Fortemps crest on it by Count Edmont's steward, and Edmont specifically says that they identify them as a ward of House Fortemps. This item never shows up again though (nor does it even appear in the Key Items list), but would again be presumed to be what the player shows to the gate guard outside the Gates of Judgement when entering the city after this point.