Yes.
The actual rules for this are oddly worded, but the notation is pretty simple.
In general, there are two formats for "things that increase your AC." The most common format is the one attached to armor:
[your AC equals] 11 + Dex modifier
or Draconic Resilience:
your AC equals 13 + your Dexterity modifier.
Note that there is no "plus" at the beginning of the formula. These items set your AC to a specific value.
The other format is that used by shields and class features that increase your AC above its base value:
Defensive fighting style:
You gain a +1 bonus to AC.
Or a shield:
Wielding a shield increases your AC by 2.
These are phrased in terms of increasing, rather than setting your AC. They are preceded by the phrase "increases" or a plus sign.
So, it's pretty simple. You get one thing that sets your AC, and any number of things that modify it (except where restricted by other rules, like the rules stating you can only use one shield).
A fighter could very easily have an AC like this:
[15 + Dex Mod] (half-plate) [+2] (shield) [+1] (defensive fighting style)
For a total AC of 18 + Dex Mod.
But he couldn't add a chain shirt to increase his AC, because both that and the half-plate set his AC to a fixed value.
The same applies to Draconic Resilience. You can stack it with a shield, because a shield increases AC. And you can further stack on the Shield spell, giving you a (temporary) total of [13 + Dex Mod] [+2] [+5], or 20 + Dex Mod.
But you can't combine Draconic Resilience with Mage Armor, because both Mage Armor and Draconic Resilience are trying to set your AC to a fixed value.
Some things Draconic Resilience stacks with:
Some things it does not:
RAW
The rules for AC are on page 14 of the Player's Handbook, or page 9 of the Basic Rules.
The relevant part is this:
Without armor or a shield, your character’s AC equals 10 + his or her Dexterity modifier. If your character wears armor, carries a shield, or both, calculate your AC using the rules in chapter 5. Record your AC on your character sheet.
[...]
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
The rules called out above appear to be these:
Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.
If you wear light armor, you add your Dexterity modifier to the base number from your armor type to determine your Armor Class.
(and so on, for each type of armor)
The rules are sloppy here. Technically, they don't give you a calculation for AC when wielding a shield and no armor. And they never explicitly call out what is or is not a "way to calculate your AC."
With that said, we can use examples to build the intent of the designers.
It's pretty clear that you don't have to choose between a shield and plate. It's also pretty clear that a shield shouldn't set your AC to 2.
Finally, it's pretty clear that wearing a chain shirt with half plate shouldn't set your AC to 28 + 2 * Dex modifier (max 2).
It seems obvious then, that a "calculation" is something that sets your AC to a value plus other modifiers.
If something just adds on to (or increases) your AC, you continue to use whichever calculation is most favorable plus the new modifier.
There are no rules that prevent this. Furthermore, there's no in-fiction reason that would prevent a DM from having a world where this is possible.
In most fictional worlds, as in the real one, one person is the product of many bloodlines:
Public domain image (source). Click to enlarge.
A Tiefling has at least one bloodline that is derived from an infernal. A sorcerer with a draconic bloodline, similarly, has at least one bloodline that is derived from dragons. A given Tiefling sorcerer could easily have both (as well as a bunch of others that are normal and human).
Of course, a given DM's world may make this combination impossible; it might be that, in their particular setting, dragons' blood and infernal blood are antithetical somehow and any such combination is non-viable, or something else with a similar end result of making the combination impossible. DMs are given a lot of explicit latitude in the DMG to create worlds with such details that may impact certain player character–creation choices. But if your DM shrugged and proceeded, then they haven't ruled it out and neither has the game.
Best Answer
Much of the transformation specifics are left to the GM
The Changeling's Shapechanger trait states:
There is not a lot of information on the specifics of the change. It is going to be up to the GM exactly how this transformation even works and what exactly it can do.
We know it can change your appearance and voice and we get a list of some ways it can do this but is this list exhaustive or just some examples? Only the GM can say.
How I would rule regarding Draconic Resilience
I would say that Shapechanger is a physical change to how you appear, not how you physically feel, nor what you can actually do (you cannot change your statistics). Thus even if somebody tried to make themself appear to not have scales, they would, in reality, still have scales and thus they would still benefit from the Draconic Bloodline Sorcerer's features.
The Sorcerer features are from your bloodline so Shapechanger isn't going to remove them
The various Draconic features are due to the Sorcerer's bloodline and changing how you appear is not going to alter your bloodline, thus these features stick around. This is also in-line with Shapechanger not changing your statistics.
The Dragon Wings feature is a bit messier:
There are no restrictions on when you can sprout wings; it does say you can do so only while a humanoid, or anything of the sort. As such, a Changeling can sprout their wings whenever they want, even while using Shapechanger (of course, as always, a GM is free to rule otherwise).
What about using Shapechanger while you have wings? The Shapechanger trait states:
So what is a limb, well a limb is:
Thus your wings are limbs meaning that technically your new form actually must have wings and you are able to determine their appearance. Of course this is a bit problematic since a Sorcerer can retract their wings, which means you get to be things like a wingless Aarakocra (again, a GM is free to rule otherwise).