So when the State learns that you’ve moved to another State, you don’t get removed from a list, you only get added to another one - but this one is private. The list of people that can ‘vote’ without knowing it.
Yes, States do not check registration against each other.
Counties within State sometimes do.
When my son went to college in State and register there, he got some official, threatening letter about being registered twice and instruction how to resolve it.
That made me happy.
But otherwise, they may drop you, only if you have not shown yourself at the pools for many years. In some places, I would think, probably never?