No, but the ability is poorly described.
The ability as shown in the Plane Shift: Kaladesh document is not worded in the standard Fifth Edition parlance, so some things are difficult to determine. It is only described as:
a natural attack that deals 1d6 necrotic damage and restores the same
number of hit points to the Aetherborn.
No mention is made if this is a ranged weapon attack, a spell attack, a melee weapon attack or similar. This is not normal for an ability description. However, we can surmise from the fact that no words such as, "whenever an Aetherborn hits with a _________ attack, it may use its Drain Life ability", that the ability is in itself an attack. Thus its effects cannot be combined with flurry of blows, a dagger attack, an arrow attack or any other type of an attack. However, as worded, it is likely that if the Aetherborn is say, a fighter with the Extra Attack feature, that this ability could be used in place of one or both attacks, since it is described as an "attack" and not an "action".