Yes, there are two types of twins. Fraternal twins, which is what you described, would just be like having a brother and a sister born at different times. They will have similarities to the parents, but they're quite distinct and different.
On the other hand, there are identical twins, of which we have, I believe, 72 pairs right now. It's in the annual report in the national data bank. One of the safeguards that we employ here is that the fingerprints are taken at the same time as the samples collected, even though the data bank doesn't know the individual by name or identity. The fingerprint is associated through the criminal history files and acts as a safeguard.
As a scientist myself saying that no two people in the world have the same DNA profile, and then immediately finding out that we have a number of these individuals in the national data bank, you can imagine that can be a bit disconcerting. Fortunately, fingerprints are able to distinguish between identical twins, and that's actually a quality assurance that we've built into the process right at the beginning.