If a student doesn't want to give or have a social security number, how does he/she go about getting the installment plan with the new MacBook Pro? Can he/she still get the student discount if a co-signer who is not a student puts down his/her information

Payment plans are contracts. Contracts require personal information. The only way for your SSN (most people have one assigned by parents before age 1) to NOT appear on the contract is if your name does not appear on the contract and you have no legal responsibility.

You'll have to take that up with whoever loans the money to purchase the computer. You show your ID, Apple gives you discount, lender gives you money, you pay Apple. Apple doesn't loan money to purchase anything. The loan agreement is between you & the lender.