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MATL, 14 bytes

1&)"V@VX&nv@]x

Try it online!

Thanks @LuisMendo for saving a byte. Explanation:

1&) % 'Pop' the first item from the input and push it on the stack.
 " ] % Main 'for' loop, to loop over the rest of the input.
 V % Stringify previous (or first) iten from the input.
 @V % Push current number, convert to string
 X& % Intersect with stringified number already on the stack.
 nv % Count the size of the intersection, and add it to the existing list of sizes.
 @ % Push the current number again for the intersection in the next loop. 
 x % Remove the number pushed by the last loop.
 % Else the program would end with the result on the second instead of the first position in the stack