Can stranded wire handle the same high voltage as solid core?
No, you'll need the proper electric fence extension wire

The problem with a standard UF or equivalent cable is that the insulation is only rated for 600V, far less than the several thousand volts that an electric fence energizer puts out.  As a result, you'll need a specialized electric fence extension wire and matching splices that are rated for such high voltages to replace the damaged wire.  Using the wrong stuff will cause the wire insulation to break down and fail, leading to your fence grounding out.