My Latest Password

Question

-What is your password?- my niece asks me.
-It is a four digit number.
-I know that.
-It is divisible by precisely three primes.
-Tell me more.
-It has at least one common divisor greater than 1 with precisely eight of the other 23 4-digit numbers that can be formed with those very same digits.
-More.
-I was alive in that particular year.
A few hours later:
-Now I know!
What is my password?

WhatsUp · Accepted Answer

EDIT: with extra information provided by the OP, now we could say that the answer is

and this also spoils a lower bound of the age of the OP, assuming he's not lying (:
Original answer:

I don't know what I'm missing, but here's what I got.
Clarifications:

I assume that "the other 23 4-digit numbers" implies that the number consists of four different non-zero digits.
I assume that "at least one common divisor" means "one common divisor greater than $1$".

With these assumptions, I got:

And I don't see any information coming from the sentences of the niece (i.e. "I know that" and "Tell me more" don't give extra information here, as far as I can tell).
Thus I'm confused at this point.
Some analysis here:

My Latest Password

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