Yes, this sounds weird. You can come up with a lot of different kinds of students and stretch it to say they aren't technically in schools ("a martial arts studio isn't a school!" Or the abstract route: "I'm a student of life!"). But the underlying truth is that "School Students" isn't a specific enough category to actually give you any sensible amount of information. You don't know what kind of students they are, how old they are, why you need to know they're in the car...
Others have mentioned that it could be because if it only said "Students" you would assume the car was being driven by a student driver. Well... For one thing, I'm still not sure the car isn't being driven by a student driver. What reason do I have to believe that "School Students" can't be a shortened version of "Driving School Students"? There clearly isn't much room left on the sign. Also, even if you do assume that is what it means, I'm still not sure what the makers of the sign want you to do with that information. "There are students of an undetermined type in this car, but they're not student drivers!" What am I going to do differently with this extremely vague piece of information? I don't know if these are preschoolers, college students, kindergarteners... Literally the only thing you know, under this assumption, is that they aren't student drivers. Well, if the car hadn't had a sign at all, I'd probably have assumed that it statistically wasn't being driven by a student driver. So what is this sign telling me that actually helps anything?
Assuming the only logical intent anyone's come up with, which is that the car has young children inside and they are urging you to be careful, I don't think the sign accomplishes that goal. "Elementary School Students" would probably be too hard to read, especially while moving on the road. But you could easily go with other messages, like "Children Inside!" or "Watch Out For Kids!". Or honestly, you could go the most easily-recognizable route and just write "School Bus" on the sign. Yes, it is clearly a car and not a bus. But if a car says "School Bus" on a school-bus-yellow sign, I'm going to assume there are kids in a car and it's being used like a bus. It's not really a huge leap.
So, I agree with you that this is a weird and ineffectual way to get across...well...whatever message it is they think they're getting across. Because clearly we can't agree on what it's supposed to mean. I'm not 100% sure it's a tautology, but in the general sense I do agree it's unneeded. If it simply said "Students", the same (vague and unuseful) message would be conveyed. So it's definitely not necessary in this situation, at least.
On both sides of the pond, school as a noun can refer to an institution for educating children (e.g. primary school), a unit or faculty of a university (University of Cambridge School of Technology), or a specialized training institution (e.g. dance school).
On top of this, in North American English, any formal education can be referred to as school. While the pre-kindergarten level is called pre-school or, at early ages, nursery school, American parents might answer yes if asked if their toddler was in school (though they may specify well, preschool, not school school).
School also encompasses undergraduate and postgraduate degree programs. ODO marks this usage as "informal," but I find that to be an overstatement. To pick a few examples from Google searches, Stanford uses the term school year in an official release; the Ph.D. admissions FAQ for Wash U answers whether someone can "keep my current job and go to school at the same time"; but there are also countless official biographies and obituaries which refer to someone's achievements while in school or at school referring to their university and postgraduate years, not to mention idiomatic phrases like working your way through school or school spirit in the language.
Moreover, this usage has been around for a while; while it isn't the easiest thing to search for, I found in the May 11, 1868 Oneida Circular in Google Books
[We] bought a house in New Haven, that we might send others to school at Yale.
And if you object that the line refers to one of the schools at Yale, the 1899 Report of the federal Commissioner of Education refers to an instructor
schooled at Yale, Chautauqua and other summer schools
and if you object that this is a verbal usage, by 1909, the influential humor magazine Puck could offer this synopsis of a life:
The New England Conscience was born in the Back Bay, went to school at Harvard in the early days, lived in Concord, and died in the New Hampshire legislature.
In British English, school is not used to refer to universities or graduate instruction.
Similarly, Canadians and Americans use student for anyone enrolled in an educational institution, whereas on the other side of the pond, it is restricted to university students; those in primary and secondary schools would be pupils. See "Pupil" or "Student", what is the correct use? for coverage of that difference; pupil is far less common than student here.
Best Answer
An "Ed School" is a department within a university or a stand-alone college that focuses on preparing teachers for their profession; also known as "Normal School," "Teacher's College" or "Teacher-Training College." http://www.memidex.com/normal-school