The Achronim discuss your question, and much of the discussion is cited in the excellent Sefer "Peros Te'enah" on Shekalim.

The MAGEN AVRAHAM (OC 470:2) rules that if a mother fasted once on Ta'anis Bechoros (Erev Pesach) on behalf of her firstborn son, she must continue to fast on that day each year, "because it becomes like a Neder." He proves this from our Sugya in Shekalim, that the father must continue to give a Shekel on behalf of his son. It is clear from the Magen Avraham that he understands that this obligation is because the father's conduct is like a Neder. Accordingly, one would have to do Hataras Nedarim if he wanted to stop. Other Acharonim, though, say that in this case Hataras Nedarim does not help (see Mekor Chaim by the Chavos Yair, OC 694); it seems that they argue with the Magen Avraham and maintain that the obligation is not because of a Neder. (Alternatively, it is a Nidrei Mitzvah/Hanhagah Tovah; once a person has begun to do a Mitzvah voluntarily they should not stop. The Tashbatz (Chelek 3 #45) brings our Mishnah as proof that if a person who is exempt from a payment that the Torah mandates and voluntarily obligates himself and begins paying, he is obliged to continue paying and cannot exempt himself again. So even as a monetary obligation he would be obliged to continue giving.)

Another practical ramification, besides whether Hataras Nedarim will work, is whether the son or other family members inherit the obligation when the father dies. If it is because of a Neder, then the obligation is only upon the father. If, though, it is some intrinsicly binding element when one gives a Shekel on behalf of his son, perhaps the family inherits that obligation if the father dies. The BARTENURA writes that the son inherits the obligation, while the TOSFOS YOM TOV argues and says that he does not.

(It is possible that the Bartenura is following his own opinion, that anyone under 20 years of age is considered a "Katan" and exempt from Machatzis ha'Shekel. Since he is nevertheless over the age of 13, it is possible for him to become obligated to give the Machatzis ha'Shekel, either because he did not disavow his father's donation on his behalf, or because the obligation is on the household and not the individual. According to the Tosfos YomTov ibid. and others, that a Katan with regard to Machatzis ha'SHekel is a child less than 13 years of age, it is obvious that there is no way that the obligation can apply to the child until he grows up. -MK)

The ROKE'ACH (333) writes that the father must continue giving the Shekel on behalf of his son "in order to be Mechanech him in Mitzvos." What he means is not clear, because how does the obligation of Chinuch obligate him to continue giving once he already gave? Perhaps it is to teach the child that once he becomes of age and must give a Shekel, he must always give it, every year.

Y. Shaw