Not true. Rings have been caught doing that but only after multiple years of voting. For example, several elections past--and no I don't have a source as this was back in the 2000's--a group of people (party irrelevant) were doing this between Ohio and Arizona. The group had all registered to vote at an address in both states. The states sent the ballots to those addresses. The local residents would mail the out-of-state ballots to the parties in the other state who would fill them out, mail them back, then the local residents would turn them in.
They only got caught because someone finally noticed that like a dozen ballots were going to the same address and raised a question about that.
So, you can vote multiple times if it is in different states because they don't cross check between each other.
If on the other hand no voter ID is required, and you have a list of voters (usually available to political parties and organizations) you can determine if anyone on it has moved out of state, died, etc., and have an operative vote for them instead. There'd be no way to know who did it once the act was completed, if somehow it was caught which isn't likely to begin with.
Then there's ballot harvesting...