Using Moq, how do I set up a method call with an input parameter as an object with expected property values?

You can use Verify.

Examples:

Verify that Add was never called with an UserMetaData with FirstName!= "FirstName1":

storageManager.Verify(e => e.Add(It.Is<UserMetaData>(d => d.FirstName!="FirstName1")), Times.Never());

Verify that Add was called at least once with an UserMetaData with FirstName== "FirstName1":

storageManager.Verify(e => e.Add(It.Is<UserMetaData>(d => d.FirstName=="FirstName1")), Times.AtLeastOnce());

Verify that Add was called exactly once with FirstName == "Firstname1" and LastName == "LastName2":

storageManager.Setup(e => e.Add(It.Is<UserMetaData>(data => data.FirstName == "FirstName1"
                                                         && data.LastName  == "LastName2")));

...

storageManager.VerifyAll();