In steady flow, the streamlines can't be tangential, because being tangential means that they overlap, even though this is only at a single point like B.
If two fluid particles (A and C) arrived at B from different streamlines with the same velocity then they would not know which streamline to leave on, because they do not have any memory. They only have position and momentum, and these are the same for both particles.
If streamlines overlap it would also mean that the fluid was infinitely compressible, because the fluid in between the two streamlines AA' and CC' has been squeezed into a gap with zero width. If the fluid is compressible the streamlines do not necessarily remain parallel at all times, but there is always a finite distance between them.
The fluid is continuous : there are no gaps between particles. So the streamlines cannot overlap by having particles from streamlines AA' and CC' arrive at B at slightly different times, with a slight delay in between them. Both streamlines are made up of a continuous line of particles with no gaps inbetween.
In an unsteady flow streamlines change from one instant to another. Therefore it is possible that at different instants of time two different streamlines could pass through the same point in space. However even in unsteady flow, at a given instant in time only one streamline can pass through a given point in space.
Source : comments by Chester Miller and Deep.