Her self-intro on the EN server says yes, they’re both Emden:

I am the Swan of the East, also known as Emden, the Dresden-class light cruiser. "And I am Emden of the Königsberg class – member of the High Seas Fleet. We are Emden. Remember our name.

I don’t know if I’ll remember their name but I’ll know their tits again when I see it haha