From Insights to the Daily Daf by the Kollel Iyun HaDaf of Yerushalayim (link):
QUESTIONS: The Mishnah [Beitzah 36b] mentions the act of Kidushin [betrothal] in its list of voluntary acts (“Reshus”) that are prohibited on Shabbos and Yom Tov. The Gemara asks why the Mishnah calls Kidushin a “Reshus” if marriage is a Mitzvah. The Gemara answers that the Mishnah refers to a case in which one is already married and has children.
RASHI explains that the Mitzvah which one fulfills by getting married is the Mitzvah of Piryah v’Rivyah, having children. Other than the Mitzvah of Piryah v’Rivyah, there is no inherent Mitzvah in the act of getting married. This explains why the Gemara mentions that the case of the Mishnah is not only where one already has a wife, but he already has children as well.
The ROSH in Kesuvos (1:12) writes that the reason why no blessing (Birkas ha’Mitzvah) is recited for the act of Kidushin is because there is no actual Mitzvah of Kidushin. The only reason why one is obligated to get married is to fulfill the Mitzvah of Piryah v’Rivyah. The words of the Rosh are consistent with the Gemara and Rashi here [Beitzah 36b]. However, the Rosh adds that one has the option to fulfill the Mitzvah of Piryah v’Rivyah without getting married at all, by merely taking a Pilegesh (concubine).
(a) According to the Rosh, why does the Gemara here say that the reason why the Mishnah does not count Kidushin as a Mitzvah is because it refers to a case in which one already has a wife and children? The Gemara should answer that there is no Mitzvah to marry a wife because one could fulfill his obligation of Piryah v’Rivyah by taking a Pilegesh.
(b) The RAMBAM (Hilchos Ishus 1:2) writes that marriage is a Mitzvas Aseh [positive commandment]. The Rambam apparently maintains that every time a man marries a woman he fulfills a Mitzvah. How does the Rambam understand the Gemara here [Beitzah 36b] which says that one who is already married and has children does not fulfill a Mitzvah by getting married?
(a) Apparently, since it is uncommon for a woman to agree to become a man’s Pilegesh (since she has no guarantee that the man will take care of her and her children, as she receives no Kesuvah and there is no Kidushin), the Gemara considers Kidushin a Mitzvah because in practice one is unlikely to fulfill Piryah v’Rivyah by taking a Pilegesh. He has no choice but to get married with Kidushin.
The Rosh in Kesuvos writes that no blessing is recited for the act of Kidushin because it is not a Mitzvah, as one could fulfill the Mitzvah of Piryah v’Rivyah by taking a Pilegesh. Even though it is unlikely that one will find a woman to be his Pilegesh, since there exists the possibility of fulfilling Piryah v’Rivyah without Kidushin, Kidushin is not considered an intrinsic part of the Mitzvah of Piryah v’Rivyah and thus no blessing is recited for it.
(b) The MAGID MISHNEH cites RABEINU AVRAHAM BEN HA’RAMBAM who was asked a similar question concerning the opinion of the Rambam. He answered that the Rambam does not mean that the act of Kidushin [betrothal] is a Mitzvah, but rather that the act of Nisu’in [marriage] is a Mitzvah (which is expressed in the wording of the Rambam in his list of Mitzvos at the beginning of Hilchos Ishus). When the Rambam writes that Kidushin is a Mitzvah he means that it is the beginning of the fulfillment of the Mitzvah of Nisu’in. Here, too, the Gemara does not consider Kidushin a Mitzvah because without Nisu’in the Mitzvah is not completed.
Alternatively, perhaps the Rambam does not mean that it is a Mitzvas Aseh per se to marry a wife with Kidushin. Rather, he means that it is an Isur Aseh (a prohibition that results from a positive commandment) to take a woman without Kidushin. (The Rambam rules that taking a Pilegesh is prohibited; see Hilchos Melachim 4:4.) The Magid Mishneh mentions this possibility later in Hilchos Ishus (1:4). Accordingly, there is no actual Mitzvah in the act of Kidushin, even according to the Rambam.