writersoul,
I once asked a geneticist I know about this (who is himself a Levi). The Kohen gene is a unique marker that is theorized to have began with Aharon HaKohen (some 80% of Kohanim have it). The geneticist told me that that there is a Levi gene as well, however only about 50% of Leviim have it (which is why it is trumpeted like the kohen gene is). The geneticist refuses to get tested himself because he wants to be able to rely on his chazaka.
The geneticist speculated that more people falsely claimed to be Leviim because there weren’t any ramifications beyond getting aliyos.