Essay writing is not limited to the US, or to humanities subjects. Both my education (Nature Science and Computer Science in the UK) and the course I currently teach on (Biology, also in the UK) made extensive use of the essay form in both formative and summative assessment. It does appear to be a more common form in Anglo-Saxon derived cultures, although it originated in France. Its also more common in less quantitative subjects: while the Computer Science part of my degree did require at least one essay to be assessed at some point, most assessments weren't essay based, the opposite is true of the Natural Science part, which had many essays and a small number of problem based exams. Humanities subjects are generally entirely essay based.
Your definition: "a short written thesis that is supposed to make an argument for or against something" is not too far off, although I'd say that it's not strictly speaking necessary for it to be for or against something. Generally an essay takes some topic, position, or statement, and examines it from as many different angles as possible, before coming to to some sort of conclusion on the topic. The structure thesis-antithesis-synthesis is common, that is, when asked a direct question, or to pass judgement on a statement, the most common conclusion is that the answer is neither yes nor no.
It has come to mean, more or less, any medium length (say 1000 words to 10,000 words) piece of writing on a coherent topic that is not a report of new research.
It is used often because a good essay demonstrates many things we would want from university graduates: they must have a good grasp of the underlying facts of the topic; they must understand the context in which those facts sit; a good essay requires the student to show good, fact based judgement; they must be able to write well; and its open-ended nature offers an optional opportunity to demonstrate mastery of material and argument beyond that taught in class or on the syllabus, including the ability to invent new arguments and theories.
Critics of the essay form argue that either it's bad because it requires all these things, rather than testing each one separately, or that most essay questions really only demand a recitation of the basic facts, along with a cookie cutter structure that makes it sound like an argument.
An example of an essay in Chemical Engineering (not a very good one as I'm not a Chemical Engineer) might be:
"'Click chemistry has revolutionized Synthetic Chemistry'. Discuss"