I'm afraid so - usually, people who work casual hours eg week to week are paid weekly e.g. for catering companies covering events. However, they may be paid monthly if they work for example for a local authority run establishment.
A contract can be verbal or written, but I would advise you son to push to receive a written contact of employment.
Does your son's payslip include holiday pay (it should be paid weekly pro rata if he is casual)? This will be worth checking as it would otherwise be stated in his contract of employment.
Whether employees are students is irrelevant. All employees will be paid under the same rules.