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first attested 1730s, but probably much older. The notion is the suit of clothes one was born in, i.e., no clothes at all. Cf. Middle English mother naked "naked as the day one was born;" Middle Dutch moeder naect, German mutternackt.
To be “in one's birthday suit” is to be completely naked (as people are at birth).
What one is wearing when completely nude; the state of nakedness: Naked as a jaybird, with nothing on over his birthday suit (1730s+)