The above picture holds the answer. So, why should maximum voltage occur when the coil is in-line with the lines of magnetic field (as shown).
Remember, the formula for induced voltage is proportional to rate of change of flux "cut" by the coil. This is an instantaneous quantity and not something that is "accumulated" over one rotation.
When the coil is in the position shown, there are no flux lines "cut" but one instant afterwards it is indeed "cutting" lines of flux so. the rate of change of flux lines cut is significant in this transitory area. Its rate rises from zero to some value dependant on the new angle of the coil and the speed of rotation.
Now consider what the rate of change of flux lines cut is when the coil is repositioned by 90 degrees (vertical to the picture shown). The maximum number of flux lines are passing through the coil but there are the same number of flux lines being cut slightly before and slightly after and therefore, the rate of change is actually zero.
A few further thoughts.
I can see how this might be confusing because if the magnetic field (from N to S) were produced from an open iron core and coil fed from an AC voltage, the maximum induced voltage to the rotatable coil would be when that coil is in the vertical position. This possibly seems contrary to the spinning coil and magnet shown in the picture.