Is it possible to align a magnetic dipole completely in the direction of field?

Great question! And the solution lies in realising that magnetisation is a macroscopic quantity. A dipole as you correctly pointed out can’t maximally align itself to the external field (due to non-commuting nature of angular momentum components).

Say the external field is in z direction. Then we will get an angular momentum state with the z component to be maximum (still less that the total magnitude). The remaining components x and y are equiprobable.

So when we add all the dipole moments, the average points in the direction of the external field.

that dipole completely aligns itself in the direction of field?

The individual dipoles don’t (can’t) align completely. That would imply we know the angular momentum vector completely, an impossibility quantum mechanically. However what does align in the direction of the field is the average dipole moment. And it is this quantity that gives rise to intrinsic magnetic field.