Find the "Recursive Size" of a List

Python, 42 bytes

f=lambda x:x*0==[]and len(x)+sum(map(f,x))

For a non-list, output 0. For a list, output its length plus the sum of the recursive outputs for its elements.

Lists fall above numbers and below strings in the Python 2 ordering, requiring []<=x<''. Instead, we check x*0==[], whereas the result of 0 for a number or '' for a string.

JavaScript (ES6), 39 37 bytes

Saved 2 bytes thanks to @edc65

f=a=>a.map&&a.map(x=>a-=~f(x),a=0)&&a

Jelly, 8 bytes

߀-ŒḊ?‘S

Try it online!

How it works

߀-ŒḊ?‘S  Main link. Argument: x

   ŒḊ?    If x has non-zero depth:
߀          Recursively map the main link over its elements.
  -         Else, yield -1.
      ‘   Increment all integers in the result.
       S  Compute the sum of the result.
          If x is an array, incrementing before adding is equivalent to computing
          the sum of the elements and the length.
          If x is an integer/character, incrementing -1 yields 0, as desired.