Two identical charges travelling parallel with same velocity (thus, relatively at rest) will experience Coulomb repulsion. However, if considered from rest frame of reference, they represent two currents travelling in same direction. Thus, they should get attracted.
What will actually happen in this situation?
In the frame of reference where the two charges are stationary you will see only the electric force, which is repulsive.
In the frame of reference where the two charges are moving you will see both the electric force, which is repulsive, and an attractive magnetic force (the electric force will dominate, but both forces are there).
The net force will be repulsive and will satisfy the relativistic force equation in both frames of reference.
This is all a consequence of special relativity.
As another answer points out, when you have steady currents in wires you get different answers. The wires in that case have overall neutral charges so the electric force is not a factor. The magnetic force, which results from the moving charges, is important though and so you get the wires moving towards each other.
The wires that you describe consist of electrons moving through a solid, such that there is no net charge. If a current is running through the wires, there is no reference frame without a current: in the frame that the electrons are still, the protons are moving. The wires will undergo a force that is proportional to the currents and inversely proportional to the separation; attractive if currents have the same sign.
The point charges you describe have a net charge. The point charges will experience a force that is proportional to their charges and inversely proportional to their separation squared; repulsive if charges have the same sign.
There is no inconsistency in the theory.
