Seems like it has to do with marrying someone who is a foreigner - Chinese military personnel cannot just marry anyone and have to have approval for whom they marry - to prevent marrying someone who will be a hindrance to military. but as far as I can tell there is no difference in the certificate itself - they use the same one so not sure why the drama infers there is a separate one - unless the approval process is what it is referring to - could just be a translation thing?

In the US, one marriage certificate suffices - then you have to present that to the various military departments to get your ID Card and have your name added to your husband's benefits, etc.