The changeling from the Wayfinder's Guide to Eberron can duplicate someone else's appearance (and voice), quite accurately
The version of the changeling published in the Wayfinder's Guide to Eberron (the current version of which also appears in Unearthed Arcana: Eberron Races) has the Change Appearance trait:
As an action, you can transform your appearance or revert to your natural form. You can’t duplicate the appearance of a creature you’ve never seen, and you revert to your natural form if you die.
You decide what you look like, including your height, weight, facial features, the sound of your voice, coloration, hair length, sex, and any other distinguishing characteristics. You can make yourself appear as a member of another race, though none of your game statistics change. You also can’t appear as a creature of a different size than you, and your basic shape stays the same; if you’re bipedal, you can’t use this trait to become quadrupedal, for instance. Your clothing and other equipment don’t change in appearance, size, or shape to match your new form, requiring you to keep a few extra outfits on hand to make the most compelling disguise possible.
Even to the most astute observers, your ruse is usually indiscernible. If you rouse suspicion, or if a wary creature suspects something is amiss, you have advantage on any Charisma (Deception) check you make to avoid detection.
There are some limitations to the ability: you can't change size (so you can't imitate Small or Large/Huge creatures), you can't change your number of limbs or overall shape, and your equipment doesn't change. And you can't turn into something you've never seen before, so you wouldn't be able to turn into a drow if you've never seen one before, or turn into the mayor of the town whom you've heard of but never actually seen. (A DM could reasonably interpret seeing a detailed image/depiction of one as "seeing" it as well.)
Otherwise, though, you are able to totally customize your appearance. The last paragraph suggests that you can look "realistic" fairly accurately; though it doesn't talk precisely about mimicking a specific individual, it can be assumed that that is what you are shapechanging into (rather than "an average example of that creature", as the shapechange spell specifies). If someone suspects that something is up, you have advantage on a Deception check made to avoid detection.
And, of course, the trait doesn't grant any ability to imitate that person other than the Change Appearance trait. You still may need to match their clothing/equipment to successfully pull off a disguise; matching someone's appearance is worth much less if your clothing clearly gives away that you're not who you claim.
Note: The Wayfinder's Guide to Eberron is a "living document", and is planned to be updated following playtesting and feedback. This trait and others may change in the future.