Kid cannot move into his apartment until after the 15th, since the unit is occupied until the 8th...yet both the kid leaving and the kid moving in (my kid) are being charged full months rent.

wtf? This sounds illegal to me. True it's not a ton of money but renting the same spot out to two people?

Can't speak to your kid's lease, but if you look back at what you/he signed, my guess is that it will probably say that the lease term begins 8/15/22 and will run to 8/8/23 (or in some cases, 7/31/23), but that they've divided the payments into 12 (or 13) equal installments. So big picture, your payment due in August may be a full amount, but it equates to 1/12 of the lease term, not necessarily the month of August.

With the massive outflow and inflow of students on the same timing, I think structuring leases this way is pretty common so they have 1-2 weeks to prep the place

What Danny described is how most do it. The lease may be a week or two short of a calendar year but basically the contract describes 12 equal installments due by specific dates each month for the total period covered by the lease.

Right; and those dates are oftentimes still the 1st of the month, helping lend the appearance of corresponding to a specific month even though they technically don't.