Now, I might just be stupid or not have seen it but, I’ve been looking and saw that draconic bloodline sorcerers get unarmored defense for “draconic bloodline”. But, do Dragonborns get this for being Dragonborn? Like I said, it’s a simple question but I can’t find the answer and my Dragonborn player wants to know because he heard that our sorcerer gets it, but he doesn’t.
Dragonborn don't get it by default
Said specifically, it is not a feature granted by choosing Dragonborn as your race when building a character. If it was it would be listed in the race's description. The two special features Dragonborn get are Breath Weapon and Damage Resistance (and the Draconic Ancestry which determines the damage type for both).
However, if a player would like to gain Unarmored Defence, one of the Dragonborn racial feats is called Dragon Hide (XGtE, p. 74). In part, the feat grants an AC of 13 + Dexterity modifier if you aren't wearing armour. This is identical to the AC provided by the Draconic Bloodline sorcerous origin's Draconic Resilience feature.
No, races and classes are different things and there is no linkage between the two
As a point of clarification, Dragonic Bloodline Sorcerers do not get Unarmored Defense. They get something very similar:
The Draconic Bloodline Sorcerer's Draconic Resilience reads:
At 1st level, your hit point maximum increases by 1 and increases by 1 again whenever you gain a level in this class. Additionally, parts of your skin are covered by a thin sheen of dragon-like scales. When you aren't wearing armor, your AC equals 13 + your Dexterity modifier.
And the Monk class's Unarmed Defense reads:
At 1st level, while you are wearing no armor and are not wielding a shield, your AC equals 10 + Dexterity modifier + your Wisdom modifier.
In the specific case of Dragonborn and Draconic Bloodline Sorcerers, there are rules- and lore-based explanations for why a Dragonborn does not automatically inherit the Sorcerer's Draconic Bloodline feature.
Rule-wise, Draconic Resiliance is a class feature that is granted by the Sorcerer class (Draconic Bloodline specialization).
The Dragonborn race has a racial feature which affords damage resistance to various types of damage based on the ancestral color/metal of the draconic ancestry.
Class features are granted by classes and racial traits are granted by races. The two are distinct and unconnected. (Save for a couple of cases where a race opens up options for class specializations but does not grant them unless they are chosen, eg Battlerager for Barbarian and Bladesinger for Wizard).
Lore-wise, while a Dragonborn is a specific race which has "a" draconic bloodline, they do not necessarily have a bloodline imbued with the specific magical properties that allow them to be a sorcerer (and thereby granted the Draconic Resilience feature)
At the same time, a dragon (not to be confused with the -born race) may shapeshift into a humanoid (and thereby procreate with humanoids):
Change Shape. The dragon magically polymorphs into a humanoid or beast that has a challenge rating no higher than its own, or back into its true form.
In procreating with a humanoid, the dragon's magic is imbued into the humanoid bloodline and this magic may manifest in the sorcerous feature that Draconic Resilience for some descendants (who would be Sorcerers).