Easiest explanation is that it’s just poorly conceived marketing copy and they’re not actually cross-checking ballots and addresses at all.
Parties can check their registered voters in most states, and counties (which are usually responsible for election administration) have all the data about your voting history, current residence, current election, etc. This is for fraud prevention (or would be if voter ID were enforced).
There’s no reason for paranoia here. It’s all public record. Maybe the recording was not particularly good, because a lot of times these recorded GOTV messages sound like the machine generated appointment call from your doctor’s office...at best.