In many systems as they are portrayed, a doppelganger and other similar shapeshifters of fluid form are given the Common language as a base and often no full telepathy (although they can detect surface thoughts in some worlds). But if the creature possesses the ability to change so much about their body why should they be constrained by somebody's else language, spoken in a standardized audible range?
With ability to transform, the doppelganger language is doppelganger themselves, they 'speak' by changing their body or, more often, the parts of their body, to convey a meaning in the way other people create specific tremblings in air or draw certain lines to communicate. Early DG language was, probably, quite literal – change to a wolf-creature if you 'speak' about wolves – but later it became both more symbolic and much more complex, including shifts in colour, texture, scents, temperature, audible and visual range capabilities to a degree, as well as shape, form and appearance, using the whole world as their language components. With no need to rely purely on the audible component to 'speak', and the natural inclination for doppelgangers to infiltrate, I think it was, at least initially, a silent language, relying on visual (and, in close proximity, textural) changes more than on audible ones, but even without a sound the DG language at its height could be incredibly byzantine.
In structural sense, the language can be seen as logographic (pictographic in early stages), with difference that each 'written'/'spoken' part of it is the cascade of changes the goppelganger themselves assumes. It probably also has several forms – at least one that is 'common people' version and is more like a quick sign language or huntspeech with only a few bodyparts and minimal changes involved, and another of highly cultured function where the whole paragraph could be condensed to a single change, and poems could be 'recited' with a single movement; the recitations of novels and sagas in this form are grand theatrics performances. If doppelganger society grows past 'oral' traditions (which, I think, would be much stronger for them than for cultures with just audible language) and come to a need to preserve the information externally, DG might have a simplified 'static' written form for the first type, and some highly ornamented multi-coloured-ink calligraphy 'static' written form for the second type, which still won't convey the language whole complexity and beauty, given how many more layers 'live' version might have.
|Depending how much the setting allows them to change, red and white flowers could be doppelgangers too; photo from Equal Wedding Japan|
Doppelganger language is possible for the outsider to learn, but impossible to 'speak' without some kind of ability to either transform the "speaker's" own body fluidly or, at least, have an active illusion that mimics such transformations (such as 'Change Self' spell). Spells such as 'Comprehend Languages' and 'Tongues' won't translate 'live' DG as the language of change isn't isn't actually written, and whatever audible ('spoken') components it has are meaningless without the rest of the changes adding to them, so translating them by magic would be like translating a written word with about 90% of its strokes missing; in its complexity, to observing non-shapeshifters, it has more similarities to a code, and this is just when doppelgangers aren't trying to intentionally 'speak' in code, or poetic, or making a double entendre jokes. Written forms are more rudimentary than 'live' forms, so even with spells the translations from written forms will lose whatever subtlety they have.
The danger for DG as to a language is in its own complexity; if the city comes under a strong outsider cultural influence, (as it often happens in the real life) younger doppelgangers might adopt the other language as their native one because it is so much easier to learn and also gives them more social mobility with the other, now dominant culture, while DG language might be seen as outdated, inefficient, pompous or, depending on a new regime, dangerous to exhibit. Simple telepathy is even more efficient, so in the settings where doppelgangers have telepathy, such DG language won't develop due to the lack of need.
On another hand, because it is rare and difficult to understand by magic, it might be used specifically to convey secret messages, similar to Native Americans code-talkers in WW2.
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On a more general thought, to function the DG city needs some kind of stable infrastructure but with everybody being able to mimic anybody else (nothing, technically, prevents a doppelganger to assume an innate shape of another doppelganger) the importance of external, preferably unforgeable symbols for important city positions is bigger than for a regular society. Assuming that doppelgangers are still individuals, with their ability to change almost everything about themselves, if they don't have no innate capabilities to recognize each other under such change (*) the masks of the office can become literally more important than the people who is currently inhabiting them; 'becoming a mask', i.e. assuming a certain specific mindset required to fit into a certain governing ability is, probably, on a job description for most of position of importance for DG city, and whatever the everyday life in such city could be, the official structure could be seen as just one endless theatrical performance by the outsiders.
(*) and if they do, it will take just one captive and/or tortured and/or brainwashed doppelganger to create a cadre of a 'drug-sensing dogs' to find all the others of their people in the vicinity, so they probably don't have such ability; 'True Seeing' spell discovers doppelgangers but it is high end spell of limited duration (and expensive to make into a multitude of items) and so not as efficient.