Sum my Fibonaccified divisors!

Jelly, 7 bytes

ÆḞÆDÆḞS

Try it online!

Explanation:

ÆḞÆDÆḞS Main link (Arguments: z)
ÆḞ      zth Fibonacci number's
  ÆD                           divisors'
    ÆḞ                                   Fibonacci numbers'
      S                                                     sum

Mathematica, 29 bytes

f=Fibonacci;f@#~DivisorSum~f&

Mathematica Simplified, 14 bytes

Fi@#~Div9`~Fi&

Oh well, this ended up being identical to @MartinEnder's solution...