Yes, I think all secondary school-aged youngsters should have their own bank account, especially those over the age of 12. It's pretty important for children to learn how to manage their money, how to budget and how to save up for things that are otherwise unaffordable from their income.
Children need to learn those vital life skills before they reach adulthood and leave home. Personally, I think money-management and budgeting should be a compulsory subject in schools.
Both of my children had full financial autonomy from the age of 14. We paid them a monthly allowance, of one twelfth of what we reasonably expected that we would otherwise spend on them (we worked out how much we would spend on clothes, outings, treats, entertainment etc in a year and divided it by 12. We didn't include food that would be eaten as meals in the family home, but we did include sweets, snacks and meals out).
They soon learned that it's easy to feel "rich" at the beginning of the month when your money is paid in, but if you spend it all in the first week there is no more until the next month, so when friends want you to go to the cinema with them or to order a pizza for a sleepover etc, you have to say no because you can't afford it until the next month!