The Bird of Hermes isn't anything, it's a metaphor Alucard uses to describe himself. Hermes, the messenger god in Greek mythology, represents Alucard himself, a servant to the Hellsing family. The second part, "eating my wings to make me tame," represents how Alucard must restrain himself due to being the Hellsings' servant.
[ In the sea without lees]
Standeth the bird of Hermes
Eating his wings variable
And maketh himself yet full stable
When all his feathers be from him gone
He standeth still here as a stone
Here is now both white and red
And all so the stone to quicken the dead
All and some without fable
Both hard and soft and malleable
Understand now well and right
And thank you God of this sight
The bird of Hermes is my name eating my wings to make me tame.
[edited to show the origin of the quote and that op is wrong.] "The Bird of Hermes" as well as the above qoute comes from the Ripley Scroll, a real thing in our world. The Ripley Scroll is an "alchemical text" and relates a a great number of things including the "Serpent of Arabia" (I assume it to mean the Devil or something simlar) and ends with the Philosopher's Stone. This is likely being used as a trope alluding to why alucard is seemingly immortal and invincible (unlike other vampires) and why its written on his coffin, which is his true resting place.
One more interpretation comes after the fact that Hermes,was not only the messenger of the gods but also the one responsible to take the dead(or rather their souls) to the underworld.Birds in greek mythology are often the form that gods take when they interact with the mortals,or are signs/helpers of the gods.Alucard is the bird of hermes,an agent of death and therefore a helper to the "soul companion" (Hermes nickname of sorts).