So the field at the centre of one loop is now the field from that loop plus an amount from the other loop that is slightly less than the field at the centre of the other loop.
So the field along the axis of a solenoid is clearly going to be different from the field at the centre of a loop with n turns.
You could imagine taking your loop of N turns and separating out each turn along the axis by some distance d, then you'd have a solenoid with 1/d turns per unit length.
The field at the centre of the middle loop would then be some complicated sum of the fields from all the other loops.
I must admit it isn't obvious to me how this sums up to the simple expression for the field in a solenoid, but presumably it does.