Emma was already at the printers when Jane Austen, in London nursing her brother Henry through an illness, was informed by his physician, who was also one of the Prince Regent's physicians (I quote her nephew James Edward Austen-Leigh's Memoir) "that the Prince was a great admirer of her novels; that he read them often, and kept a set in every one of his residences; that he himself therefore had thought it right to inform his Royal Highness that Miss Austen was staying in London, and that the Prince had desired Mr. Clarke, the librarian of Carlton House, to wait upon her. The next day Mr. Clarke made his appearance, and invited her to Carlton House, saying that he had the Prince's instructions to show her the library and other apartments, and to pay her every possible attention. The invitation was of course accepted, and during the visit to Carleton House Mr. Clarke declared himself commissioned to say that if Miss Austen had any other novel forthcoming she was at liberty to dedicate it to the Prince. Accordingly such a dedication was immediately prefixed to `Emma,' which was at that time in the press".

So no, Emma was not written for the Prince Regent and any apparent references to him are either coincidence or were there before the dedication was - well, though couched as a request it was pretty much ordered. Jane Austen made use of it, though. The John Murray Archive now in the National Library of Scotland in Edinburgh includes a letter from JA to Mr Murray, her publisher, complaining of delays at the printers and saying