The Vampiric Bite ability of the Dhampir race can be used as an attack, so RAW the Dhampir Echo Knight's Echo can bite. But would this be able to Empower the Echo Knight?
The Echo never actually bites anything; you bite as if from its space
The Manifest Echo feature states:
[...] When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo's space. You make this choice for each attack. [...]
Notably, it is any attack you make, so when using the Dhampir bite to attack a creature, even if the bite originates from the echo's space, it is still you making the attack. Thus all the benefits of the bite apply to the fighter themself and not their echo.
Both Manifest Echo and Unleash Incarnation specify you make the attack...
I would assume you're referring more to the ambiguous [Unleash Incarnation] Feature of the Echo Knight when asking this question, rather than the [Manifest Echo] effect. [Unleash Incarnation] states:
[...] You can heighten your echo's fury. Whenever you take the Attack action, you can make one additional melee attack from the echo's position.
While this attack is exclusively performed from the Echo's location, you are still the one performing the attack. Therefore, it should empower the PC as if they are the one performing the action. Which can lead to some very powerful combos if you turn your bite into a magic weapon in some way, like:
- Tasha's Otherworldly Guise
- Magic Weapon
- Hex Warrior
Unfortunately, Ki-Empowered Strikes specifies unarmed attacks, so RAW it cannot make your simple weapon Vampiric Bite magic. Sidetracked, I know, but I figured if you're trying to use the Vampiric Bite in this way, you'd like some other ideas.