Or does coinbase not actually send bitcoin but simply adjust balances internally within their website?
^ This.
They are not sending Bitcoin from a address to another. They are simply just changing your balance info on their database to a lower or higher amount when you transfer money. They actually only send Bitcoin through the "real" blockchain when you want to withdraw to another wallet outside their system.
Does that mean they have full control over my money with them and if they want they can change my balance? This is alarming as there should be a mechanism so that they cannot change my balance. You may say they will only do it when instructed by me (by transferring btc to another wallet) but it leaves the accounts prone to human error.
Didn't you know that they have full access on your money? Every online wallet has that access and also if they want, they can change your balance (but in reality no company does that if account isn't blocked for serious reasons). It was mentioned, they are just changing your balance, imagine xapo offers such service also, free send between xapo users. When sending bitcoin, it doesn't matters if wallets are same or not as you asked, you always have to pay fees as others do.