The oldest senses of this word, which arrived in English from French in the 14th century, are all to do with something belonging specifically to someone - at first property, then personal qualities. From there, it developed the sense of "individual, particular", but it seems not to have developed the nuance of "odd" until the 18th century.
So it is important not to read this in, when we encounter such uses as Hamlet's "single and peculiar life" (Hamlet, III.iii.11), which means no more than the life of a private individual, or the captured Lucius talking about "my peculiar care", meaning "his own personal care" (Cymbeline, V.v.83).
The word turns up three times in Othello. Iago talks about following Othello for his own "peculiar end" (I.i.61) and later persuades him that many "nightly lie in those unproper beds Which they dare swear peculiar"
(IV.i.69), while Desdemona (III.iii.79) comments to Othello on the absurdity of seeing her request as if she were asking him "to do a peculiar profit To your own person". In all these cases, the meaning is no more than "particular, private, personal".