No source connects it to Oscar Wilde, who died in 1900. The earliest exact match Quote Investigator found is an anonymous line at the end of a Usenet post in December 1999, already in quotation marks. In September 2000 it was printed, unattributed, in an advertising supplement for the Menards store chain. The credit to Wilde came later.
What it means
There is no point imitating someone else: every other identity is occupied, so the only one open to you is your own. It is a joke with a serious point about authenticity.
Where it comes from
Quote Investigator notes that the line is not in The Wit & Wisdom of Oscar Wilde, the large collection compiled by the quotation expert Ralph Keyes. Wilde did write often about identity and masks, but none of his remarks matches this one. The nearest earlier thought Quote Investigator found is Thomas Merton's, in a 1967 essay: there is “very little chance of my being anybody else.”
After the 1999 post the line was soon credited to a person named Gilbert Perreira, and it appeared in a Menards advertising supplement in September 2000 among other unattributed inspirational sayings. Wilde's name became attached as it spread online. As Quote Investigator concludes, the true originator is unknown.