Broken: 2.80*, 2.81*,2.82, 2.83, 2.90
Worked: None, as there wasn't a multi-object edit mode before 2.8x.
Description of the error
Given two selected objects with linked object data, they can't go together into edit mode if either has a modifier with the option to display in edit mode(turned on) and if "On Cage" option is turned off. I don't know if this is intentional or not, I believe it's a bug because then we're only able to see transformations from edit mode propagating to the other linked objects after going back to object mode. In normal circunstances, the linked objects go together into edit mode, even when only one of them is selected, plus we're also able to see the instances being updated in real time when editing the mesh. This is not what's happening when there's a modifier.
Now it gets more weird. This I'm sure is not intentional. If the modifier's "On Cage" option is checked, we're able to see the updates from changes made in edit mode on all instances in real time, but also the effect of that modifier shows on every linked object, even those without modifiers! Note that in 2.80 and 2.81, this happens even with On Cage turned off. The only exception to this part is the wave modifier even with "On Cage" turned on it won't go into edit mode with other instances.
Exact steps for others to reproduce the error
duplicate default cube
select both and ctrl + L then link object data
with both selected go into edit mode
add a solidify modifier, see that all instances except the active object leave edit mode.
move a vertex of the mesh, the instance won't update unless you go back to object mode.
now turn on "On Cage" option of the modifier, you will see that in edit mode all instances show transformations in real time,
but now they look as if they had the active objects modifier.