After doing this, if I select the original one and try to apply the modifier, I cannot because the menu has been disabled. How can I apply it? I want it to be applied to both of them equally.
Modifiers (and Transforms) cannot be applied to an object that shares mesh data-block with another user. And Alt+D creates just that: linked copies.
Either “make single-user”
OR
duplicate with Shift+D
OR
(apparently) Object > Convert > Mesh applies all modifiers to Active Object and updates shared data-block across all users (what? it can do that? I didn’t know that Oo)
Seems like that Convert → Mesh is the only way to apply the modifiers and to keep the two linked at the same time.
While you can’t apply modifiers to object with linked data, nothing stops you from:
1 unlinking data in selected object (make single user)
2 applying modifier on selected object
3 relinking data from selected object to all with previously same data
4 copying modifiers from selected to all objects to remove double modifier effect
If that sounds like a lot of keypresses and menu searching for something like that logically should be one-two clicks then check out modifier list addon:
It also makes working on modifiers with object gizmos (empties) a lot easier:
Just note that by default ui is set to old style list (hence name of addon) so change it to stack.
