By the rules
Dragonborn have the dragonblood subtype, which is defined as
If a race possesses the dragonblood subtype, it has a strong affinity to dragons—which means that spells, effects, powers, and abilities that affect or target dragons also affect it. The subtype qualifies a creature to use magic items normally only usable by dragons, and qualifies the creature to take feats that have the subtype as a prerequisite.
The dragonblood subtype also makes creatures subject to harmful effects that affect dragons. The dragonblood subtype does not confer the dragon type or any traits associated with that type. For instance, it does not give a creature frightful presence.
Dragons automatically qualify for any classes, prestige classes, racial substitution levels, feats, powers, or spells that require the dragonblood subtype. Races presented in this book that have the dragonblood subtype include dragonborn, spellscale, kobold, and draconic creatures. Should a creature acquire the dragon type, it loses the dragonblood subtype.
(Races of the Dragon pg. 4; also, Dragon Magic pg. 4)
So dragonborn is considered to have a “strong affinity for dragons,” to the extent that “spells, effects, powers, and abilities that affect or target dragons also affect it,” and it can also “use magic items normally only usable by dragons.” Furthermore, the dragonblood subtype is required for a number of feats in Races of the Dragon and Dragon Magic.
Since they lack the Dragon type, dragonborn do not qualify for many feats, or classes, reserved for dragons. They certainly aren’t “true dragons” as required by a select few special features.
By characters in the game
Dragonborn obviously have draconic aspects; anyone can see that. They equally-obviously are not your typical dragons, since they have a generally-humanoid shape, lack a tail, and many lack wings. They are also much, much smaller than adult dragons. So people aren’t going to have the same reaction to them that they would to a dragon, particularly a full-grown one.
As for dragons themselves, those are notoriously proud and arrogant; they are not likely at all to consider a dragonborn an equal, though the good ones will likely approve of their devotion to Bahamut.