There is no customer list. The only thing Blizzard could do is ban anyone using the same IP address as someone on Turtle WoW. However since NAT is widespread in many countries and many people don’t have their own IPv4 address this would result in an extremely high number of false positives. Not to mention that multiple people could be sharing the same internet connection. Besides there’s no reason to do that. Someone who also plays on the official server is paying for a subscription. Banning that person now would just mean less revenue for Blizzard.
Presumably Turtle WoW also had payment details. If those match up with the IP of someone playing WoW, Blizzard has a very good case for action.
And yes there's a good reason: playing on a pirate server directly and explicitly violates the EULA. Blizzard bans people for violating the EULA all the time.