If my wife and I are married filing jointly, do we both claim our daughter as a dependent, or is there a way only of us does?

If you are married and filing jointly, then you are filing on the same tax return.  Claiming your daughter as a dependent on your joint return means you are both claiming her.  There is no designation as to which one is claiming her, you are filing a joint return.  If you file separately, then only one of you can claim her as a dependent.

So what if one parent signs in the portal and wants paper check and the other signs in and wants direct deposit