Shouldn't it have both an IMEI and ESN/MEID number?
Nope, despite (apparently) having the same components, the 4S is still carrier-specific. 4S models for GSM/HSPA carriers will have IMEIs, while those for CDMA carriers will have MEIDs.
And unlocked iPhones, even factory-unlocked models, come with an IMEI but no MEID, and cannot be activated on CDMA networks, either.
Also: Sprint's long-standing policy has been to only activate phones with ESNs or MEIDs in their inventory. If the MEID isn't listed with Sprint as being "one of theirs," they won't activate it.
By and large the same is true of Verizon. They
claim to permit non Verizon phones on their network per a FCC mandate, but the phone has to be deemed as "compatible" with their network and undergo a rigorous, draconian testing process, even if it's supposed to be an identical model to what they're already selling. Basically, it's a load of red tape designed to strongly discourage anyone to who would want to try.
Think about it: if any iPhone 4S really
could work on any network, then Apple wouldn't have separate inventories for Sprint, AT&T or Verizon, with stores running out of each one separately. There would just be one big pile of 4S models, and you'd be asked what carrier you wanted at the point of activation. And that would actually be easier for them too. Then they wouldn't have problems where the Verizon or AT&T models sold out and the Sprint models didn't because no one happened to want to be a Sprint customer on a specific day.
Unfortunately, this something the carriers DON'T want. Having truly unlocked muitl-mode phones out there would mean that the subsidy-contract model would make less sense to people. Finding ways to discourage customers from being able to freely move from carrier to carrier is mutually beneficial to them, and so policies are put in place in the name of "network security and integrity" to bar people from using their unlocked phones to do just that. And so, while each carrier-specific iPhone 4S may come out of the same factory and have the same components, and technically ARE capable of working on any network, artificial barriers put up by the carriers still make these devices "incompatible" with each other's networks, except maybe for roaming purposes.
Bottom line: the OP was given the wrong type of iPhone 4S. he needs to get another replacement, this time for the right carrier.
