Problem:

Calculating the percentage of the characters from each string in column B that appear within the corresponding string in column A.

Solution:

Use the SUM, ISNUMBER, FIND, MID, ROW, INDIRECT, and LEN functions as shown in the following Array Formula:
{=SUM(--ISNUMBER(FIND(MID(B2,ROW(INDIRECT(\"1:\"&LEN(B2))),1),A2)))/LEN(B2)}

To apply Array formula:
Select the cell, press