I can speak generally about higher education admissions standards to universities in the 17th-18th centuries, which were comparable between European and American universities. Prospective students had to demonstrate working knowledge of Latin and classical texts, which were assessed either through written exams or oral presentations. Because universities in both America and Europe were often established in conjunction with Christian churches (first Catholic churches in Europe, then Protestant churches in America) a strong command of Christian theology as well as 'godly character' would have also been necessary. I cannot say how universities would assess prospective students on the latter, as it would be very subjective. Prospective students were likely admitted if their family consisted of prominent community members who had both the money to pay a child's college tuition and enough of a positive reputation to count for 'godliness.' Admissions standards did not include assessment on other subjects, like mathematics or the sciences, until around the 19th century.

Just because these admission standards seem simple does not mean that anyone could enroll at a university. To know Latin, the classics, and important theological texts, a student would need to be educated extensively. This would mean you'd need to be at least somewhat wealthy (to afford the schooling, since not everyone could receive this level of education) and male (women's education primarily consisted of homemaking skills, etiquette, music, and sometimes French, if you were an English or American woman from a well-to-do family. Universities also did not admit women.) Though students would not be barred from attending if they were of a lower social class, it would be very unlikely that a student of this social status would be able to have access to the resources (and connections) to get themselves admitted. A less wealthy male student whose father or relatives was a pastor might have had access to the early education needed to meet admissions standards, in which case they may have attended with financial support from others or through working while in school. Finally, this also does not account for the fact that wealthy people in America regularly traveled to Europe for higher education between the 17th and 19th centuries, especially for training in medicine at Paris and Edinburgh.

Overall, I would say that the admission system in the 17th-18th centuries does not look at all like it does today, and it was both less structured and less accessible to the vast majority of people.

This historical account of Ivy League universities' admission standards may be of interest to you.