In Hilchos Channukah (OC Siman 675 Sif 1 — See also there Mishneh Brurah Sif-Katan 3.) We learn that if one lit Channukah candles (for the mitzvah) from the day before and they stayed lit until the following day still he must extinguish the candle(s) and re-light it now for the sake of the mitzvah (of ner channukah) since each day is a mitzvah in and of itself.
What about in regards to Shabbos. If someone lit Shabbos candles Erev Shabbos Parshas Chayeh Sarah (for example) and they stayed lit until Erev Shabbos Parshas Toldos and would easily burn well into the night, is this enough? Or must new candles be lit in honor of this Shabbos? The main reasons for Shabbos candles is for Shalom Bayis in order one may see around the house. As well the enjoyment one has by eating by the candles. Seemingly these 2 things are fulfilled with the candles that remained lit from the previous Shabbos.
This question is the subject of a dispute among authorities. In practice, the candles must be put out so that the berachah can be made upon re-lighting them.
The Mishnah Berurah, based on the Rema and the Magen Avraham, rules that the light must be lit especially in honor of (this) Shabbos.
The reason for this is that they maintain that the idea of “hadlakah oshe mitzvah” (the mitzvah is performed in the action of lighting) applies to Shabbos candles, just as it does to Chanukah candles.
As proof to this the Mishnah Berurah mentions the proof of the Gemara to the halachah of “hadlaka oseh mitzvah” from the blessing over the candles, which refers to “kindling the Chanukah lights.” The same wording is used for Shabbos candles, indicating that even for Shabbos candles, the mitzvah is the actual act of kindling.
See Rema (263:10), Mishnah Berurah (263:48) and Magen Avraham (22) citing from the Mordechai (Shabbos 273). See also Mishnah Berurah in the Laws of Chanukah (675:1).
However, others authorities write that concerning Shabbos candles the mitzvah is not the act of lighting, but rather having the light, and according to this there is apparently no obligation to extinguish the candles and re-light them in honor of Shabbos.
[As the question notes, this opinion stands to reason based on the rationales of the Shabbos candles, which is to ensure that there is light in the room, though there is also a rationale of “kevod Shabbos,” and for this reason it can be argued that the candles must be lit especially for (this) Shabbos.]
This opinion is stated explicitly by the Taz (675:1), and the Eliyah Rabbah (5765:1) questions his opinion from the aforementioned Gemara: Surely the blessing we recite over the candles is proof that the essence of the mitzvah is the act of lighting.
Yet, a number of rishonim back up the opinion of the Taz, as cited by the Shulchan Aruch Ha-Rav (263, K.A. 3, citing from Tosafos, the Ran, and the Rosh (Shabbos 2:18) writes explicitly that one fulfills the mitzvah with a candle that was lit from before, and there is no need to extinguish the Shabbos candles and light them again.
Concerning the proof from the Gemara, see Pnei Yehoshua (Shabbos 23a) who defers the proof.
However, even according to this opinion, one must put out the candles in order to make a berachah upon re-lighting them for Shabbos.