They won't see race, or barely, but they will see what you make. So if you do something to alter your appearance then they will see that. So the races of the Dwarves (that is, until we get to stuff like Derro or Duergar), are the materials they generally work with.
The Dwarves we are familiar with are big on steel as they have a lot of it but other Dwarves tend to think of that as their specialist ethnic material. (Handy, hard, but how do you stop it rusting?)
Dwarves of Gold and Ivory.
Make elaborate mail of glass beads, which helps to keep them cool in their hot climates. Super curly hair means they tend to think of is as a kind of construction material for the head, forming beards and haircuts into asymmetric wedges, or sometimes three dimensional models of their home city or mountain.
Tend to break down laughing whilst trying to relate a joke they told a week ago, (these Dwarves laugh) its so funny in their memory that they can't get all the way through it without breaking down.
Really want to tell you about their family, in depth, for hours. Family is huge and has collectively done and experienced some of everything so no topic so strange that they do not have a handy homily about it.
Tend to announce that they are going to speak to you before, or as, they speak to you. i.e. sits down and says "I will speak to you now."
Iron is a currency to them and they wear thick torcs of raw iron around their arms and waists.
Most are very black and they smear fine gold dust all over themselves as a kind of makeup or war-paid, the dust catches in the folds of their skin and makes them seem to glow like bronze. Other dwarves think looking like this is pretty cool and are slightly annoyed they can't really carry it off.
Dwarves of Jade and Wine.
These have wild black bristles that stick straight out. Very distinct mustaches. Tend to wear a lot of jade, including a kind of closely arranged mail of jade pieces that jingles softly as they walk, a distinct sound for each clan.
These Dwarves are named so as they are one of the few kinds of Dwarf that actively enjoys wine more than Beer or Spirits. (All Dwarves will drink wine if available, but these will actively seek it out.)
Brutal calligraphy duels with two-handed calligraphy brushes, if a dwarf dies they can still win the duel if they leave behind the superior poem on the paper floor where it is fought. The loser will always accept the superiority of the poem if it is better, even if they killed their enemy. In fact they will make a point of mentioning it in the future. "I bested Shan Lun in the Poets-cage, but only his flesh is dead, his words will burn through time when I am dead." Then they recite the poem.
These Dwarves are so friendly and luxurious when drunk, and so often drunk, that they are the only Dwarves that sometimes seem lazy. (They aren't really, but can sometimes give that impression.)
They like monkeys, are friends with them and even have a monkey god. (Very unusual.)
Dwarves of Obsidian and Bone
These very fierce Dwarves have even less metal available to them than most. Amazingly, they ritually shave themselves and tattoo elaborate abstract beard designs onto their faces and chests (presumably slowly expanding over the body during their life). They wear a kind of elaborate gorget/mask of obsidian shards which takes the place of a beard and which is fucking terrifying.
Their weapons are obsidian and hardwood, as well as other organic materials such as bone. They are always on the lookout for new kinds of bone and will raise incredible animals over centuries, simply to harvest the bone when they die.
In their homelands they are called 'Cave-Lords' and actually rule as an upper caste from very well-developed terraced farmlands on the shoulders of high mountains. They travel beneath the earth through the caves that are common to that part of the world and generally do not bother low-living peoples much, or at all, unless monsters threaten, in which case they issue forth with great violence
(Our (Western-standard) Dwarves call themselves the 'Dwarves of Stone and Steel', but really everyone likes stone, so when not present they tend to be called the Dwarves of Steel and Beer. Although most Dwarves are pretty fond of beer too.)
I was going to go for an Indian-Standard set but its hard to get a strong material signature for Indian crafts. Is there any material or set of materials that brings that culture strongly to mind? (Obviously dwarves with Bakh Nakh are a must so they have to have those. I was thinking of making them the only dwarves that actively use a wide range of weapons instead of just impact-damage stuff. Sikhs are so close to the cultural dwarf-zone that its hard to not just re-make them shorter.)