FIV is in the family of lentiviruses, and these viruses persist once a cat is infected - they don't go away. So the presence of FIV antibodies shows infection. More false positives occur in low risk groups. There can also be false negative results. A negative test can be repeated in 3-4 months to confirm it is a true negative.
Kittens get antibodies to many diseases, in the mother's milk.
Kittens younger than 6 months may have FIV antibodies gotten from the mother, without having FIV themselves. Of course, keep the cat strictly indoors to limit exposure to infection. Get regular vaccinations, as long as the cat is in the latent stage. Check with your vet about vaccinations for a cat who is symptomatic. When you see any signs of illness, go to the vet - early treatment can head off many problems.