Prior to the creation and rediscovery of the Primarchs, the Legiones Astartes were predominately known by their legion number rather than a particular name. In the case of the Blood Angels, they were the IXth Legion (also known as the Revenant Legion), and in the case of the Dark Angels, they were the Ist Legion.
Now, for the Blood Angels, the name comes directly from their Primarch, Sanguinius. Sanguinius was notable for having massive white wings not unlike those of traditionally-depicted angels, and overall resembled a typical angel in his appearance. After being rediscovered by the Emperor and given command of the IXth Legion following battle on the world of Teghar Pentaurus, Sanguinius declared of his legion that "even though a darkness hangs over them, a future soaked in blood and horror, they are angels yet. Angels of Blood." And henceforth the IXth Legion was known as the Blood Angels.
The Dark Angels, on the other hand, being the Ist Legion (and for a time the only Legion), were referred to as "Angels of Death" directly by the Emperor himself. Later, after the rediscovery of their Primarch Lion El'Jonson, the Ist Legion was officially renamed the Dark Angels after an element from a Calibanite fable (Caliban being the world on which the Lion was raised) in which an army of shadow-enshrouded angels descended from the sky to aid the Calibanite people in a time of great need (which is generally how many Imperial citizens would come to view the Adeptus Astartes as well, though the Calibanite myth likely predates the creation of the Astartes by a considerable margin).
In regards to the similarities between their Legion/Chapter iconography, the use of wings is present across the Imperium, and at first glance one might think that there was a running connection between them. Several sigils including those of the Adeptus Astartes (Space Marines), the Astra Militarum (Imperial Guard), and the Imperium itself bear the motif of wings, derived from the Aquila emblem that symbolizes the Imperium at large. The Imperialis, a decoration in the form of a winged human skull, likewise features this motif and is prominent on Space Marine armor. However, it appears that their similarities are mere coincidence: The Dark Angels' badge of a winged sword was a pre-existing emblem of a Calibanite sect of monastic knights called The Order, which discovered and raised Lion El'Jonson after his arrival on Caliban, while the Blood Angels' badge of a winged drop of blood is probably a reference to Sanguinius' naming of the Legion and to the Primarch's own wings.