If everybody was issued a unique PIN, sure, but if the PINs are picked at random you need a much larger pool of numbers to make 1000 unique values likely.
If 1000 people chose a three digit PIN completely at random, the odds of any one person's choice being unique are about 36% ((999/1000)^999) - i.e., about 360 people will get a unique number, the rest will share theirs with at least one other person. In fact, it's almost certain (about 99.9%) that one of the 1000 possible PINs will be picked by five people (see https://math.stackexchange.com/a/25878). That formula estimates that there's even a 75% chance that your group of 1000 people contains one group of six who all picked the same PIN.
If you go up to five digits, for a pool of 100,000 possible PINs, then the odds that one of a thousand random PINs is unique go up to 99%, so you likely only have 10 or so people who share a PIN. Six digits, you get to 99.9%, which means about half the time 2 people have a PIN collision.
So no, capturing enough uniqueness from a fingerprint (given that you don't get to allocate the fingerprints, they get chosen for you at random) to reliably pick out one person from 1000 requires at least as much entropy as a six digit PIN - probably more.
If 1000 people chose a three digit PIN completely at random, the odds of any one person's choice being unique are about 36% ((999/1000)^999) - i.e., about 360 people will get a unique number, the rest will share theirs with at least one other person. In fact, it's almost certain (about 99.9%) that one of the 1000 possible PINs will be picked by five people (see https://math.stackexchange.com/a/25878). That formula estimates that there's even a 75% chance that your group of 1000 people contains one group of six who all picked the same PIN.
If you go up to five digits, for a pool of 100,000 possible PINs, then the odds that one of a thousand random PINs is unique go up to 99%, so you likely only have 10 or so people who share a PIN. Six digits, you get to 99.9%, which means about half the time 2 people have a PIN collision.
So no, capturing enough uniqueness from a fingerprint (given that you don't get to allocate the fingerprints, they get chosen for you at random) to reliably pick out one person from 1000 requires at least as much entropy as a six digit PIN - probably more.