Verizon message+ does not sync your own texts with iphone texts
Ask Verizon why their app does not have some feature you think it should have.
Re why you think it's strange that the recipient sees the message in both apps... what else did you expect to happen?  Verizon's Message+ does not implement a different communication channel, it's a received SMS.  Of course both are going to show it, they'd be useless if they did not.