Is the following a reduced form?
He's grown up knowing your name.
If yes, what has the original form been?
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This is not a 'reduced' form.
Knowing here acts here in its participial capacity, not gerundial, so the phrase/clause knowing your name acts as an adjectival. Syntactically it is a 'secondary subject complement': a non-obligatory predicative complement of the verb which identifies or characterizes the subject. An ordinary adjective or noun phrase could play the same role:
He grew up very poor.
He grew up a beggar.
You could paraphrase
He grew up, and all the time he was growing up he knew your name.
But don't mistake this paraphrase for a 'fuller' form of the construction.
He's grown up [knowing your name].
No, it's not a reduced form.
You specifically asked what the grammatical function of "knowing" is. The simple answer is that its function is head, more specifically head of the bracketed gerund-participial clause.
I take the subordinate clause to be an adjunct, since it can be freely omitted, though I'm not sure what semantic category it belongs in.