Yes, you should receive a 1099-NEC for that work. Since the student is a business, he is required to to issue you a 1099-NEC, if the amount was over $600. That said, given the circumstances, I don't think you're going to get into trouble if you don't give him your SS#. His problem is that the IRS is aggressive about taking business deductions for which a 1099-NEC was not issued.
Since you file a Schedule C, you should get an EIN for your business, so that you don't have to give out your SS#.
You do not have to actually have employees to get an EIN. How to get an Employer Identification Number (EIN):
https://irs.taxid-us.com/?keyword=get%20a%20federal%20tax%20id&source=Google&adAccount=460-350-8849&...
As an aside, can he really deduct the tuition as a business expense?
Probably. W-2 employees can no longer deduction education as a job expense, but the self employed still can. The general rule is: Educational expenses to improve your current job skills are deductible. But learning a new job is not. A personal development course is probably in a gray area.
