In the early days there wasn't a formal process for .edu approval. There was a bit of netiquette that educational institutions should register under .edu, nonprofits under .org, network providers under .net, and companies under .com.
That started being enforced for .edu in 2001, but previous registrants were grandfathered in. The registrar did manage to evict a bunch of the questionable .edus in 2003, by stepping up enforcement of technical requirements, such as the requirement to have accurate whois information, and to be responsive via the whois-registered email address. But presumably academia.edu survived that purge. I can't actually think of any other prominent for-profit company that still retains an .edu. The registrar also now prohibits transferring them, so you can't buy one even if some others have survived (though maybe you could lease it).
That started being enforced for .edu in 2001, but previous registrants were grandfathered in. The registrar did manage to evict a bunch of the questionable .edus in 2003, by stepping up enforcement of technical requirements, such as the requirement to have accurate whois information, and to be responsive via the whois-registered email address. But presumably academia.edu survived that purge. I can't actually think of any other prominent for-profit company that still retains an .edu. The registrar also now prohibits transferring them, so you can't buy one even if some others have survived (though maybe you could lease it).