What, exactly, is a "co-pilot"?
Does "the co-pilot" have any meaning?

Yes, Merriam-Webster says:


  co- prefix
  
  :having a usually lesser share in duty or responsibility: alternate: deputy • copilot


and


  copilot
  
  :a qualified pilot who assists or relieves the pilot but is not in command


while dictionary.com says:


  variant of com- before a vowel, h, and gn: coadjutor; cohabit; cognate. The prefix co-, now productively forms new words from bases beginning with any sound ( co-conspirator; co-manage; coseismic), sometimes with the derived sense “auxiliary, subsidiary” ( coenzyme; copilot), and, in mathematics and astronomy, with the sense “complement” ( codeclination).


Both sources mention / hint at the co-pilot not being the highest in the command hierarchy.

Is co-pilot ever a rank?

Yes, the co-pilot is the second-in-command (SIC), right after the captain (even if the co-pilot has more hours than the captain—it does happen). If the captain ate the fish, the co-pilot becomes pilot-in-command (PIC).

Does the term appear in aviation regulations?

It's very rare (nowadays at least), usually second-in-command will be used. Here's a mention of co-pilot in an FAA advisory circular (not a regulation per se).


  The Mentor Pilot. A mentor pilot is a professional aviator acting as a coach, co-pilot,
  teacher, and role model, sharing experience and expertise in the complex environments in which
  the EA500 will operate.


Is it used in crew management policies?

Not really. The terms used are pilot-not-flying (PNF) or pilot monitoring (PM), and pilot flying (PF). Because the captain and co-pilot can be either, they usually swap roles. The captain can be the PF on the out-bound leg, and PM on the return flight, for example. But the captain remains the PIC even when the co-pilot is the PF.

Is there indeed any standardised or formal use of it within the industry?

Popularity varies by location / culture / era. First officer (F/O) is also a valid alternative. Searching for jobs on Google, copilot returns ~9 million results, while F/O returns 30 million.

Using Google's Ngram Viewer could be skewed due to the F/O's usage in maritime environments.



What does ICAO say? (My addition.)

It's not bothered really:


  What is a multi-crew aeroplane?
  
  It is an aeroplane that requires a flight crew of at least two pilots. One of them is the pilot-in-command (the captain) and the other is the co-pilot (or first officer).




Related:


What are common airline crew titles and positions?
When does a flight require a copilot?
What are the roles of the co-pilot in commercial aviation?
Can two captains fly an airliner?
What is a Second Officer?