Husk, 8 bytes

ḟȯ=L1LUT

Try it online!

ḟ # first element that satisfies
 ȯ=L1L # length (of string) equals length of input (array), of
 U # longest prefixes of all unique characters of
 T # transpose of input

Husk, 8 bytes

ḟȯ=L1LUT

Try it online!

Different approach to the previous Husk answer . Gets the longest prefix with all-unique characters from each index position, and then selects the first that is the same length as the number of input strings.

ḟ # first element that satisfies
 ȯ=L1L # length (of string) equals length of input (array), of
 U # longest prefixes of all unique characters of
 T # transpose of input