The Regulations require only the first payor to be listed on the 1098-E. That does not mean co-borrowers cannot claim the interest if they paid it. As you point out, this is similar to the treatment of home mortgage interest.

26 CFR § 1.6050S-3 - Information reporting for payments of interest on qualified education loans.

(2) Payor. Payor means the individual who is carried on the books and records of the payee as the borrower on a qualified education loan. If there are multiple borrowers, the principal borrower on the payee's books and records is treated as the payor for purposes of section 6050S and this section.