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This you see as the north pole forming near the north pole of the solenoid. Like poles repel, unlike poles attract is kind of a misleading analogy(I find it strange that I realized this after reading your question). What actually is happening is that the tiny magnetic dipoles inside the magnetic material(a bar of iron, for example) align with the direction of the magnetic field.
And that's where the names come from. The 'North pole' of any magnet is the one that would be attracted to the Earth's North magnetic pole. In that sense, they are just labels that identify the opposite magnetic force produced at each end of the magnet. The Earth's magnetic field also has poles, North and South.