During the period in which Austen lived and wrote there were seven distinct classes in England. Austen finished writing Pride and Prejudice in 1798. She finished writing Emma in 1815. The same class structure spanned all these years.
The highest class was comprised of royalty and lords, ministers of State (e.g., Chancellor) and titled nobility down to, but not including, baronet. The second class comprised baronets and knights and wealthy country gentlemen and anyone else with a large income. Darcy and Bingley both belonged to this class. The third class holds educated professionals like doctors, the clergy, barristers, banker, merchants, large manufacturers. Mr. Collins and Mr.Weston were in this class.
Mr. Woodhouse belongs to the second class as a wealthy country gentleman. Mr. Bennet belongs to the second class for the same reason (though his wealth is dwindling). Therefore, both Pride and Prejudice and Emma are told from upper class points of view. The second class is one of the two upper classes; this is why it is acceptable for nobility (1st class) to marry gentlemen's daughters (2nd class) since both are of the upper classes.
It is a common misconception that the Bennet family is of the middle class, but this notion is not at all substantiated in the novel. The Gardiners represent the middle class as they made moderate money in trade. At home, Elizabeth is shown with her mother and sisters and none of them are working in the house or in a garden. Mrs. Bennet boasts that she doesn't have to teach her daughters to cook because she can afford a cook!: “Mrs. Bennet … assured him with some asperity that they were very well able to keep a good cook.” Mr. Bennet is shown either with his wife and daughters, “The astonishment of the ladies was just what he wished,” as in the opening chapters, or in his library: “in his library he had been always sure of leisure and tranquillity.” They are described as an upper class family of (dwindling) wealth and leisure.
The social class differences shown between the novels relate to extra characters, not to the Woodhouses nor to the Bennets. The highest class individuals the Woodhouses know are Mr. Knightly and Frank Churchill. Mr. Knightley is important but has no title so is in the second class along with the Woodhouses. Frank Churchill was adopted by the Churchill family who also have no titles so are also in the second class. Mr. Elton is a clergyman in the third class with educated professionals. The lowest class individuals, aside from the gypsies, are Harriet, farmer Robert Martin, Miss Bates, and Jane Fairfax.
The highest class individual Elizabeth knows is Lady Catherine de Borough who seems to be in the second class also, otherwise she would have a title of baroness or higher. Elizabeth also knows Colonel Fitzwilliam who is the younger son of an earl. The lowest class individuals are Wickham, Mr. Collins, and Uncle and Aunt Philips in Meryton.
You can see that the stories are told from exactly the same class, which is the second class, not the middle class, which would actually be members of the third and fourth classes. By examining the characters, e.g., Mr. Knightley and Lady de Borough, you see that Elizabeth associates with characters of higher status than Emma does. By examining lower class characters, like Harriet and Mrs. Younge, you can find the differences in social class representation in the novels.