I've been using Straight Talk set up on the Verizon towers for about 3 months - it always says "Verizon" as the service.
I suspect service will always show the name of the towers being accessed (Verizon, ATT, etc,) rather than "Straight Talk".
(Actually, I'm in a camper in a mountain rural area now, and service says "network extender" - not mine lol.)
As long as it works, don't worry.