It’s not really a question of being “formally labelled”. Some people with MS consider themselves disabled, others don’t.
As far as the workplace is concerned, yes, it is formally recognised as a disability right from diagnosis - whether or not the person is severely affected (this is partly to stop employers quickly getting rid of employees they consider are “bound” to get worse, and cause a lot of problems). But again, it’s up to the employee whether they declare things, and so claim that protection.
Studying, of course, is not quite “the workplace”, so the same legislation does not apply. However, most (All?) universities ask whether you consider yourself disabled. If you have a formal diagnosis of MS, I don’t think there’s any issue at all with answering “yes” to that question.
I certainly did a couple of years ago, when I did a short unit with the OU. They were very nice about it, and falling over themselves to help me, but none of the “special help” that might have been available was really relevant to me. Because I was doing it from home by distance learning, and there were no face-to-face sessions or exams, I didn’t have any transport or access issues, and they could potentially have provided things like speech recognition (if I’d lost ability to type), or course materials in audio format, if I had vision problems. But none of these really applied. My problems are of a more general nature, such as fatigue and pain. There was nothing practical they could offer, to help with those specifically. The one thing that might have been useful, which was extra time to complete assignments, was the one thing they couldn’t/wouldn’t do; I had the same submission deadlines as anyone else, and if I missed them, I’d have missed them, regardless of reasons. (In the event, I didn’t).
So it all depends, really, on how you are affected, and the nature of the support available. “Coming out” as disabled doesn’t necessarily mean they can magic up something that will help. If you have trouble writing, you may be entitled to the services of a scribe in lectures, or extra time in exams, and that sort of thing. But if the basic problem is: “I feel pants”, there’s not really an adjustment to deal with that.