Q:

Primes ’n’ Digits

This has no practical purpose but it could be fun to golf.
Challenge
Given a number n,

Count the amount of each digit in n and add 1 to each count
Take the prime factorization of n
Count the amount of each digit in the prime factorization of n, without including duplicate primes
Create a new list by multiplying together the respective elements of the lists from steps 1 and 3
Return the sum of that list

For example, 121 has two 1s and a 2, so you would get the following list from step 1:
0 1 2 3 4 5 6 7 8 9
1 3 2 1 1 1 1 1 1 1

The prime factorization of 121 is 112, which gives the following list for step 3:
0 1 2 3 4 5 6 7 8 9
0 2 0 0 0 0 0 0 0 0

Note how we did not count the exponent. These multiply together to get:
0 1 2 3 4 5 6 7 8 9
0 6 0 0 0 0 0 0 0 0

And the sum of this list is 6.
Test cases
1 -> 0
2 -> 2
3 -> 2
4 -> 1
5 -> 2
10 -> 2
13 -> 4
121 -> 6

Notes

Standard loopholes are forbidden.
Input and output can be in any reasonable format.
You should leave ones (or zeros for step 3) in the list for digits that did not appear in the number.
This is code-golf, so the shortest solution in bytes wins.

A:

Jelly,  18  17 bytes
-1 byte thanks to caird coinheringaahing & H.PWiz (avoid pairing the two vectors)
DF‘ċÐ€⁵
ÆfQÇæ.Ç‘$

A monadic link taking a positive integer and returning a non-negative integer.
Try it online!
How?
DF‘ċÐ€⁵ - 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
    Ð€  - 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

A:

Jelly, 16 bytes
ṾċÐ€ØD
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.
ṾċÐ€ØD     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']
 ċÐ€       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

