This will be a very likely legal victory, but a potentially hard legal path. This is wrong for a bunch of factors. The downside is there's a lot of interesting legal exploration to do here, and some "new law" has a chance to emerge (i.e. influential case law), and it would take years to litigate it all. The good news is, you only have to win on one arm of what I'm about to say, and it's so obvious I would expect a quick judgment.
Social media is a very serious commitment
If you sign up under your real name, everyone who knows you will link you. Then, they will expect you to interact - react to their posts, talk to them in private messaging, accept real-world party invitations on-platform -- and do so timely, or their feelings will be hurt and they will distance from you in-real-life.
Unless you are very careful, they will see sides of you that you never meant to expose, like your keyboard personality and your politics. You should certainly expect social media to change/damage your long-term friendships. The platforms are also engineered to be addictive. To be competent on social media is a serious time/life commitment. Quitting social media is awkward and creates social problems all its own.
The problem is, the teacher is prescribing a commitment far beyond the scope of the class. Beyond classmates, beyond class time, and effectively creates a public exposure and life commitment for the student.
Saying this is going to get me downvoted by those who are inculcated into those social media platforms, but deny how entwined they are, or see only the positives. I suspect the teacher of the class is one of them, and is oblivious to these impacts. The school may be run by a clique of these people.
Is it responsible to oblige a child into social media?
The law rules that a 14 year old is not competent to make significant and irreversible life decisions. Now, it's true lots of people just play fast and loose with social media, and that's ... up to them and their parents. But given all the stuff I mentioned in the last section, there is a material question of fact as to whether such conduct is responsible, or good parenting.
It seems to me that it is axiomatic that the parents have not only the right, but the parental duty to guide such an impactful decision. Or to be more precise, "no social media for my child" is an entirely reasonable parenting decision.
For the school to oblige the student onto social media, they are tearing that decision out of the hands of the parents. Schools do that all the time for decisions like "Shall the child be taught of the existence of the US Civil War", or "shall the child be forced to learn New Math methods." But the decision to join social media is a big one.
The Right of Privacy of the student and family
Not everyone wants to be a Kardashian; not everyone chooses to be in the public eye. The legal system has struggled with "the right of defining your own level of notoriety" for a long time. By "levels" I mean someone being forced from a private person to a social media presence; not someone seeking fame/already famous from becoming more famous (though managing the nature of your publicity is also recognized by the courts).
The problem here is that the student is being compelled into a public presence that is unwanted.
That might make sense if the crux of the coursework was to do that, i.e. if the class was an elective course about public presence in social media. However, it looks like a broad, all-in or all-out program of which social media is only a facet. And clearly, the requirement wasn't disclosed in advance.
There's also the genealogy issue. Now you're publishing information about other people, which opens other cans of worms. Not least, those others have no legal recourse against a minor; that would fall against the parents!
Obviously schools get to force some speech: they can make you do your homework! However, this is normally confined to the student-teacher relationship, or at worst the class at large. However, this is NOT on-point.
Because here, the forced-speech is public in a novel way -- on social media. It invites interactions with outsiders (another thing you don't normally have in a class, and remember, the school has no ability to curate this public response, and they may have a responsibility to curate what is presented to a 14-year-old as pat of classwork). And it involves establishment of a public persona. In that light, the question of forced speech needs to be revisited given that existing case law won't fit it.
Irreversibility of an entry to social media
One doctrine in the courts, actually codified in Europe, is "the right to be forgotten".
The child always has the option to forego social media for now and join it later. However, one cannot unring the bell; one can't simply fade and be forgotten off social media, especially if someone has done something infamous.
Consider the "Cash me ousside" girl, who self-annihilated on Dr. Phil's TV show. Most future employers would find out about this and terminate the woman on the spot, for cause. The girl was 13 years old. The mother probably made a bad decision to consent to this, but the girl (legally) had no possibility of understanding the consequences of the actions, so obviously, no means to consent.
The law simply does not recognize an idea that a 13-year-old is fully responsible and should pay the price indefinitely for youthful indiscretions. The entire juvenile justice system is built around this notion. The EU has codified it in the "Right to be forgotten". So this is a gory mess legally. As it happens, this child, has chosen to monetize the bad girl image and monetize it, but was it really a choice? What other options exist? The reputation is inescapable. The parent and Dr. Phil should've foreseen that.
Legally, this is a gory mess. Who is responsible? Will Dr. Phil or the mother assure the grown woman an income? It's practically turned into a science fiction story.
So what happens when a child in this program makes a mistake that places them in infamy, with blowforward 10, 20, 30 years into their future, again from a child's mistake? Who's liable for that? The parents for consenting? Is it really consent if the alternative is to fail the student? The school for requiring? Are the school's lawyers as good as Dr. Phil's? In things like this I often figure "whose lawyer would I rather be" - not the school's.