The GRE vocabulary is infrequent to meet in academic life. What is the purpose of learning them?

I think it helps a lot here to understand the American cultural context, and in particular the liberal arts model of education that is quite common in the US.

The GRE, and the departments and universities that care most about them, thinks of graduate training as pointing towards a career as a professor, not as a researcher. In the US, the majority of jobs as professors have teaching (at the university level) as a valued component, and in the US, students at university are supposed to take a broad range of courses rather than courses focused on a single area of study. This means that professors are also teaching an audience that isn't solely focused on their subject but rather students with a wide range of interests (at least in the ideal case - in non-ideal cases, they may be teaching students with no interests).

Even though I'm a mathematician and teach only mathematics courses, I think I am a better teacher (and academic advisor) because I know a fair amount about philosophy, religion, history, literature, economics, sociology, computer science, and so on. I'm not an expert in any of these things, and I make this clear, but I can make analogies to a broad range of ideas and use them in conversation with students.

For an American, the vocabulary section of the GRE is meant to test, not whether you can memorize the meaning of a large number of words for the GRE, but rather whether you are a widely read person who has already encountered many of these words in the course of their life and studies.

Certainly the verbal GRE scores are generally taken into account much less seriously (or not at all) when making graduate admissions decisions for non-American students, particularly when the student has no plans to remain in the US after their studies.

The point is not to "study for the GRE", but to have a good vocabulary, so that if/when other people use subtle words, subtle concepts, figurative phrases, and/or references to things outside the immediate textbook, it's intelligible. I say this as a mathematician, who often finds reason to make analogies to things more obviously part of the external world, and sometimes in terms that are not inside a small vocabulary. Thus, students, especially grad students doing more sophisticated mathematics than most undergrads, would benefit from acquaintance with a larger vocabulary (and with the referents of those words).

(Still, let me be clear, I am not a fan of the GRE itself, for a variety of reasons...)

The vocab in the GRE is a measurement of verbal reasoning. The GRE never makes any claims of being practical. It strictly wants to know how well you can do on a general test that measurements several factors related to graduate studies.

Naturally, there have been complaints about the GRE's ability to predict graduate school academic performance. However, as of now, this is a required test for admission to many graduate schools