I can confirm this issue.
The Eden callsigns do not work in multiplayer. They are only set on the server.
I have attached a mission to test this:

• One of the groups has this setGroupId ["NATO DC -"] in the groups init field. => This works in multiplayer.
• One of the groups uses the Set Callsign Module. => This does not work in multiplayer.
• The other groups use the Eden callsign settings. => This does not work in multiplayer.

I believe the suggested fix, to change the generated Eden code from setGroupID to setGroupIDGlobal, would fix these issues.

Additionally there is another critical issue, that is preventing us from using the Eden callsign effectively:
Cutting & Pasting a group resets the callsign. The same thing happens when copying a group and pasting it into another mission.
https://feedback.bistudio.com/T118543

in the this config Cfg3DEN >> Group >> AttributeCategories >> Callsign >> expression
replacing _this setGroupID [_value]; to _this setGroupIdGlobal [_value]; would be a great tweak

Thank you for reporting this issue.
We will see what we can do

I can confirm! Eden attribute works fine on dedicated server in current dev branch

Similar

problem occurs in the «Set Callsign» module function \a3\modules_f\groupmodifiers\functions\fn_moduleGroupId.sqf

Or isGlobal value can be changed to 1 as solution