Premise: Social interaction in RPGs can be influenced by what character class you play. This is obvious, after a fashion: folk treat priests differently to soldiers. Therefore, when you play a character class you can emphasise the social traits of that class or not.
A mechanical notion: the maintenance of the social traits can be put into the hands of the player, not the GM or similar. First, this is because the player needs to play the character - is the fighter a stoic, loyal, soldier or a raging, homeless, barbarian? Secondly, the mechanical aspect of remembering to keep the social traits or cues present is useful, in that the player must maintain this status as an act of will.
These very mechanics are flexible, but a character's weekly schedule might include some time set aside to the cultivation of those traits. Aside from any choices they might make regarding items: if the cleric puts on a pagan amulet, regardless of the benefits (+4 to XYZ)), even if it has no real supernatural downsides - is going to look odd.
As any aspect of a society, these are going to be differences between settings. I am going to employ the European Medievalism of Terrae Vertebrae, paired with the system I know best: The 52 Pages.
Fighter: making it clear to the world at large that you are a fighter is not hard - just carry a sword and armour. However, there are certain specialities needed to gain the 'Fighter Social Trait Bonus'. First, do not wear any armour, clothes or other accoutrements with the symbols of a king or lord or other power structure. It is acceptable to have dwarf hallmarks from the smith on your helm; it is not acceptable if the helm is covered with the war-runes of the Dwarf Kingdom of the Bronze Chasm.
Second, you may wear a symbol - but only a voided symbol: a plain, blank black shield. (A quick look at this post may be relevant for unaffiliated men-at-arms - though that is a very regulated mercenary world). This cannot be concealed.
Rogue: the mind goes instantly to gang colours or pirate flags or thieves' cant. But would one really wish to indicate that one was a Rogue? The class title is what it is and need not have any given interaction with the world of crime. Does a rogue even have a social status?
The answer (or my answer) is to say: like the fighter, but with more looseness, more swagger. It is not necessary, either to refrain from wearing symbols: it is necessary to defend yourself from those coming after you for wearing them. Likewise, there is an exhibition of successes: wear your loot! This is a distinction: a fighter is methodical; a rogue is intuitive. However, you must refrain from interaction with the more prominent criminal gangs.
Wizard: the staff is traditional. But the thing that really proofs you as a wizard is a license. The wizard must proof their abilities and that they can use them safely. Therefore, the main social cue that comforts folk around a wizard is a license, or diploma. This is generally carried in a dedicated scroll case in an accessible place on the wizards body (a belt pouch, or across the shoulder). The display of a single scroll case, for many, is a good enough indicator in and off itself of being a socially benign wizard.
Prophet: the class is called prophet, not cleric. No-one insists on you wearing liturgical vestments in the dungeon - still less a dog-collar. Therefore, I would opt for another aspect of religious practice: the tonsure. The connection to monastic life sets it aside from the secular clergy and the Church hierarchy. As above, not wearing pagan amulets is entirely necessary. Wearing a symbol of the Faith of the Eight or appropriate Saint's icon is - if for no other reason than Van Helsing vampire repellant ones.
Dwarf: don't trim the beard. Keep all hair braided carefully. Humans can grow facial hair; only dwarves care for and properly maintain a beard (or braid; I've never jumped one way or the other on female dwarves and beards).
Elves: don't conceal the ears. Terrae Vertebrae elves tend to have a link or association with a place and community in that place; those wandering will carry something to carry on the link. This is probably not a pot plant, but more a locket contraption perhaps - at least, for woodland elves. A phial of seawater for the insular elven communities.
Caprines: very much as in the link. Don't conceal the horns or the hooves.
Hereafter, Next 52 specialities.
Bard: the musical instrument must be on display. Colourful attire - at least, hat and scarf or other peripheral attire.
Militant: a mix of fighter and prophet. Don't display symbols - excepting religious ones. Tonsure or equivalent advised but not compulsory.
Mystic: as wizard and prophet - display both of license-scroll case and religous symbol. Avoid pagan amulets; tonsure or equivalent advised but not compulsory.
Mountebank: as rogue, but more colourful and flamboyant (as Bard).
Whether or not a character wishes to be known as an especially Dwarven dwarf, or wishes to conceal themselves, or simply wants to dress in a certain way is up to them. But if you want to trade on your position without any other kind of references or letters of introduction or preceding reputation, this is the way to do it.