I think that independence is used to describe a situation that is independent relative to the alternatives available to the person described. In Miss Taylor's case, which is very unlike Emma's, she has no fortune of her own and her life is maintained at the expense of a man who is basically her employer. Even though she no longer functions as Emma's governess, she earns her keep by being the companion to Emma and to Mr. Woodhouse that they want her to be.

For women in similar circumstances, whether with family or employers, marrying a respectable man she loves, one who has either inherited or made his fortune, puts her in the position of running her own household and raising her own children. In relation to the alternative of earning her keep another person's household, this offers her more independence in the sense of her having more power to direct the household, and to raise her own children rather than others' children.

She does not own a home all by herself, but if her husband is rational and good, he will treat her as more of a valued partner, and somewhat less as an employee. But IMO being a good wife (especially then, and still today) is not as different from being a good employee as some might suppose.

I think that the following exchange from Chapter 5, between Mr. Knightly and Mrs. Weston, illustrates the dynamic to which I refer.

"I should have been sorry, Mr. Knightley, to be dependent on your recommendation, had I quitted Mr. Woodhouse's family and wanted another situation; I do not think you would have spoken a good word for me to any body. I am sure you always thought me unfit for the office I held."

"Yes," said he, smiling. "You are better placed here; very fit for a wife, but not at all for a governess. But you were preparing yourself to be an excellent wife all the time you were at Hartfield. You might not give Emma such a complete education as your powers would seem to promise; but you were receiving a very good education from her, on the very material matrimonial point of submitting your own will, and doing as you were bid; and if Weston had asked me to recommend him a wife, I should certainly have named Miss Taylor."

"Thank you. There will be very little merit in making a good wife to such a man as Mr. Weston."

"Why, to own the truth, I am afraid you are rather thrown away, and that with every disposition to bear, there will be nothing to be borne.