Suspect you are giving them too much credit and they did not mean any such allusion and were not even aware of that old Nazi organization.
Also, I don't think "Joy" is a concept that either the Nazi party nor the Democrat party should be allowed to commandeer. It rings hollow when they use it in both cases.
And I think you are giving them too much benefit of the doubt. They know exactly what they are doing and saying.
They might be aware of it. Hardcore leftists know their own ideology quite well.
It was Walz who introduced this— nationally. It was known to him, as he is quite clearly a Chi-Com groomed and cultivated ultra Socialist (btw of German background like so many who settled Minnesota. The main Democrat party in MN is the Farm-Labor Democratic Party— the party of Hubert Humphrey back then and still today).
Now it’s right up there with the rebranded meaning of GAY.