The crib-drag attack applies with a different operator.
p1 = AtextencryptedByFirstPassword
p = HelloHelloHelloHelloHelloHell
c1 = hxpihlrncmwxpopfjtcgatldgdsco
p2 = ASecondEncryptedTextBySecondPassword
p = HelloHelloHelloHelloHelloHelloHelloH
c2 = hwpncuhpyqycaeskxpihicdpqvroaozwhzfk
When you $c_1 - c_2 \bmod 26$ you will have the same problem as many-time-pad. This subtraction will remove the key.
c1[i] - c2[i] = p1[i] + key[i % len(key)]- p2[i]
But this time instead of $\oplus$ we will have plus modulo 26.
c1 - c2 = azafvjqcwecflqdfowgbijsmkszmm