Technically this is a misuse. But its evolution comes not from non-native speakers making grammatical mistakes, rather it comes from (presumably) native speakers attaching new (technical) meanings to the word "verify".
The real question the service rep is asking you is:
Please verify your identity by stating your address.
The meaning here is the same meaning you said you understand: to confirm your identity. And to do that you are asked to state your address as a form of password.
In computing and security this step is called verification. And the reason it's called that is obvious: the user needs to be challenged to ensure he's not an imposter. So technically the verification is of the user, not the password. So you'll see in a lot of code and publications of computing related papers and documents related to policing and security where the word "verify" is used in the same way the rep you spoke to is using it.
Technically the usage is incorrect. But English tend to evolve to embrace commonly used language constructs. I would not be surprised if this usage would be considered normal in the future.