Frankly, there's never enough description of monsters. When I see something
like this for a doppelganger, it's no surprise that the flavour has gone out of role-playing and that so much energy has to be spent crying out for more of it in the game. There's simply nothing on the page linked to help a DM effectively run the monster or get into the monster's head.
The original description from the 1979 monster manual was not much better.
As such, we have to turn to other sources to gain insight into the creature. I can think of two in-depth examples right off, though of course there are hundreds, since making one character that looks like an existing character is an old Hollywood trope. It is effective, creepy and -- in terms of production costs -- cheap. But while the T-1000 turning into John Connor's foster parents is a familiar enough motif, there are only two examples I can think of that attempted to dig deeper into a doppelganger's motives and mental state -- and both were "heroes," not villains, while neither ever used the term "doppelganger."
Some argument can be made that Mystique wasn't exactly heroic. Her character was waltzed back and forth across the line of pragmatism so much that it eventually suffered from the necessity to satisfy the plot rather than maintaining a strict and acceptable character. Even Jennifer Lawrence got to the point where she didn't want to play the character any more, a fact that was obvious from the way she walked through the part in the last two movies (and the reduction of her make-up quality because either the actress didn't have time to sit in a chair for six hours or agreed to sign on again only if she didn't have to).
But, the question of whether a doppelganger actually likes looking like someone else, or many someone else's, is a good dramatic question. In D&D, the matter is settled because adopting another persona gets the doppelganger what it wants: power. Asking a natural shapeshifter if it gets tired of reshaping itself is something like asking me if I get tired of eating, sleeping, writing or going for a walk. Of course, yes, sometimes I do get tired of these things. I don't know what to eat, I have things to do and I don't want to go to bed yet, I don't know what to write, I don't feel like a walk today. But these things are actually so natural, and often filled with such moments of joy and satisfaction, that nothing about them are enough to make me stop. Logically, a doppelganger could no more seriously want to stop changing shape than one of us would want to stop showering to get clean.
Then again, D&D has lots of doppelgangers, who have a culture and can share thoughts with each other. There's only one Mystique. That would have to really screw up a person. How exactly it would screw up a person is hard to judge, however, which I would count as the principle reason why the character was never really pursued. We could afford to give her a few minutes of angst in every film, but then the actual plot has to get resolved, so Mystique, if you wouldn't mind transforming yourself into an army colonel now, we could maintain the film's momentum.
A better investigation took place with Odo in
Deep Space Nine, though of course now we're talking about someone who was distinctly an anti-doppelganger. Much of his unwillingness to change shape resulted from
the budget of a television show, especially given how much CGI costed in the 1990s to do relatively simple things believably. His character had to be designed around that reality, so that most of the time Odo did entire episodes where this didn't take place at all. It was an extreme case of
informed ability -- though an understandable one (not like Riker). We had to be told that Odo didn't like to change publicly, treating his shapeshifting ability as something dirty, like masturbation, that needed to be hidden in his room because it embarrassed him ... and the show established the reason for that by adding backstory that he was treated like a dancing monkey in his early experiences with humans. We can relate to this and understand it, so it worked. Saved the makers of the show lots of money.
[I'm sorry that I tend to see films and television -- particularly the latter -- in terms of how much a particular sequence, set or bit of CGI costed. One of the reasons I enjoy excessive blockbusters is that the expenditure is so obscene that I stop worrying about how much a particular effect cost, because so much of the film obviously involved thousands of people spending a year of their lives nitpicking over pixels. I can just forget about budget details. On the other hand, if I watch something like the Umbrella Academy and some vortex opens after we've just spent 35 minutes watching actors shout and mumble at each other, I am immediately struck by how much that little feature must have cost someone; and when the super-intelligent ape character appears on the scene, I feel a little sad knowing that I'm not going to see that character very much because wow, that would be expensive to have Pogo on screen constantly]
Odo worked as a character because he wasn't especially educated, and his back story did suggest someone without much imagination ... which also subverts every argument that Odo could have saved Deep Space Nine single-handedly. It is one thing to have enormous power -- it is another to have it without possessing much insight on how to solve problems. This is something every DM recognizes in the 17th level characters drenched with magic items who find themselves stymied by three levers sticking out of a wall, that must be pulled in some sort of order.
It would be hard for a DM to concoct a character as dense and three-dimensional as Odo ... and certainly impossible to do it from the paltry sum of material that we find written under a monster with Odo's powers. This is true across the monster spectrum. The books can give us details on how a monster functions, but that is really just the machinery. The actual motivation underlying the monster is hard to convey, and frankly most "monster books" don't even try. The pleasant benefit of working online, in a blog or on a wiki, is that space is not relevant; nor is a "final version," which is what the WOTC adopts anyway because surely no one has looked at their monster compendium pages since they were first put in place, probably five or six years ago. A creative soul can, like a television show that allows a character to be explored episode to episode, return to the original material and keep adding more. Details that were lacking, an explanation of motivation, a hint as to how a particular power could be used or how to subvert it, can be thrown in as it occurs to the creator and, ultimately, the user as well.
So, today's monster is the
doppelganger. I had fun working it up this morning, using notes from other sources in order to build a better picture. Found a good picture for it, too.