I'm a US native, and I have never heard of such a thing. It is almost a game for various groups and organizations to designate "days," to get publicity for their group or cause. If they send out a press release saying that some day is National Chocolate-Covered Pretzel Day, some newspapers will run stories saying "Did you know that today is National Chocolate-Covered Pretzel Day?" and perhaps they can sell a few more chocolate-covered pretzels. Since there isn't any national organization organized by, or promoting the interest of students, I doubt that there is any-- --wait, I'm wrong. There is one. I guess. https://www.nationalstudentday.com "Sponsored by the National Association of College Stores, National Student Day spotlights and celebrates students through events in campus stores across the U.S. and Canada. On October 13, 2016, get to your participating campus store and join the party!" It's one of those fake "days." They're trying to promote their business and get people into their stores. It isn't a "real" celebration, festival, or holiday. You might be able to get some free snacks, and there might be a sale price on things for your dorm room... that sort of thing.

Not that I'm aware of. Is there one on Russia? That might help us to answer.

Yes and No. Looking at this: https://en.wikipedia.org/wiki/Tatiana_Day it looks like the first university in Moscow was started on St. Tatiana's Day. Now, she's the patron saint of students and every year there's a big celebration on her day. No, we don't celebrate her day (I would highly doubt many Americans without Russian Ancestry have ever even heard of her). However, a lot of schools and colleges celebrate their founders' birthdays or the day of the founding of their school with similar celebrations (lighting a fire, having student parties, having some sort of official meeting for their school, going through some rituals, giving out awards...) if you were to go to one of those schools during their celebration, you'd probably think "Oh, this is just like Student's Day!" But it isn't national, every school would have do something different on a different day.