If we have two magnets and one is brought towards the other, the north of the former magnet is facing north of the latter. Now the other magnet is repelled which is obvious and Newton's third law is applied.
But is it applied in all frames of reference or if we replace the magnets with charges or current-carrying wires?
The solution shown on this website isn't just enough and I still don't understand. Whoever answers, please also provide another example.
Newton’s Third Law is secretly the conservation of momentum. Electromagnetism conserves momentum. However, it is possible in principle for non-negligible momentum to be stored in the electromagnetic fields, rather than transmitted to the charges and wires which we think of as being more tangible.
Your linked example is about two current elements which are perpendicular but not intersecting, and shows that one of the elements is parallel to the field produced by the other (and thus feels no $I\times B$ force), while the converse is not true. However, two isolated current elements don’t obey conservation of charge — the current has to go somewhere. Real skew currents feel an aligning torque, and then are attracted to each other as they become more parallel. The example here chooses one segment at the “fulcrum” of the aligning torque, where the associated force happens to vanish.
An analysis of the forces between free charges on skew-perpendicular trajectories, keeping track of the momentum stored in the changing electromagnetic fields, would be an interesting teaching exercise.
I am not familiar with your reference. I'm not saying they are wrong, but they are talking about the magnetic forces associated with currents.
With regard to your permanent magnets the forces they exert on one another are equal and opposite per Newton's third law. See:
Hope it helps.
