Difference between single circuit and double circuit transmission line protection systems
No, double circuit transmission lines do not have a special protection system.  Each circuit  has its own protection system like any other transmission line.  Typical protection schemes being POTT, PUTT, DCB, current differential, or regular old step distance non-pilot.

Yes, mutual coupling is an issue but it’s also an issue on many single circuit lines that just happen to share right-of-way for a portion of their distance with other line(s).  Plus, mutual coupling is really only an issue for directional ground elements that operate on zero sequence. Using negative sequence directional elements substantially alleviates the issue.  The effects of mutual coupling on reach elements (ground distance) are fairly easily accommodated as long as you properly model it in short circuit studies.  J. Lewis Blackburn covers this well in his textbook, “Protective Relaying Principles and Applications”.

A more troublesome problem with double circuit lines is the possibility of back flash after a lightning strike producing an inter circuit fault.  e.g. An AB-ground fault involving A-ph on one circuit and B-ph on the other circuit.  This can “flip out” protection logic in the line relays.