Primes ’n’ Digits

Jelly,  18  17 bytes

-1 byte thanks to caird coinheringaahing & H.PWiz (avoid pairing the two vectors)

DF‘ċЀ⁵
ÆfQÇæ.Ç‘$

A monadic link taking a positive integer and returning a non-negative integer.

Try it online!

How?

DF‘ċЀ⁵ - Link 1, digitalCount: number(s)    e.g. [13,17]
D       - to decimal list (vectorises)            [[1,3],[1,7]]
 F      - flatten                                 [1,3,1,7]
  ‘     - increment (vectorises)                  [2,4,2,8]
      ⁵ - literal ten                             10
    Ѐ  - map across              (implicit range [1,2,3,4,5,6,7,8,9,10])
   ċ    - count                                   [0,2,0,1,0,0,0,1,0,0]

ÆfQÇæ.Ç‘$ - Main link: positive integer, n   e.g. 11999
        $ - last two links as a monad:
      Ç   -   call the last link (1) as a monad   [0,2,0,0,0,0,0,0,0,3]
       ‘  -   increment (vectorises)              [1,3,1,1,1,1,1,1,1,4]
Æf        - prime factorisation                   [13,13,71]
  Q       - deduplicate                           [13,17]
   Ç      - call the last link (1) as a monad     [0,2,0,1,0,0,0,1,0,0]
    æ.    - dot product                           8

APL (Dyalog), 43 41 bytes

⎕CY'dfns'
+/×/+/¨⎕D∘.=⍕¨(⎕D,r)(∪3pco r←⎕)

Try it online!

How?

r←⎕ - input into r

3pco - prime factors

∪ - unique

⎕D,r - r prepended with 0-9

⍕¨ - format the factors and the prepended range

⎕D∘.= - cartesian comparison with every element of the string 0123456789

+/¨ - sum each row of the two tables formed

×/ - multiply the two vectors left

+/ - sum the last vector formed

Jelly, 16 bytes

ṾċЀØD
ÆfQÇ×Ç‘$S

Try it online!

Developed independently from and not exactly the same as the other Jelly solution.

Explanation

I'm gong to use 242 as an example input.

ṾċЀØD     Helper link
Ṿ          Uneval. In this case, turns it's argument into a string. 
           242Ṿ → ['2','4','2']. [2,11] → ['2', ',', '1', '1']. The ',' won't end up doing anything.
    ØD     Digits: ['0','1',...,'9']
 ċЀ       Count the occurrence of €ach digit in the result of Ṿ

ÆfQÇ×Ç‘$S  Main link. Argument 242
Æf         Prime factors that multiply to 242 → [2,11,11]
  Q        Unique elements → [2,11]
   Ç       Apply helper link to this list → [0,2,1,0,0,0,0,0,0,0]
     Ç‘$   Apply helper link to 242 then add 1 to each element → [1,1,3,1,2,1,1,1,1,1]
    ×      Multiply the two lists element-wise → [0,2,3,0,0,0,0,0,0,0]
        S  Sum of the product → 5