Simply use the Array modifier instead.
Add an empty and flip it in X and Y axis
Add Array modifier > Object Offset and target the empty
You can flip the empty either by scaling to -1 or through Object > Mirror
Mirror modifier mirrors the object's geometry in it's local space, so if you select X axis and the geometry is towards the +X direction from the origin of the object it will be mirrored to the -X direction, if the geometry is towards the -X direction from the origin, it will get mirrored to the +X direction. You can change the origin point using transform ...
Short answer, no, you cannot add modifiers to collections.
In you pic your are adding the modifier to 'Cube' as it is the selected object.
You can however copy modifiers to other objects with CTRL+L
For an array modifier, to keep all your parts relative to each other you'll need to use Object Offset and also have all the objects origins at the same point, ...
Your view is showing $Y$ and $Z$ axis (Green and Blue lines across the screen respectively), but the modifier is set to mirror only on $X$, which would be perpendicular to this view, therefore impossible to see.
You can add mirror in $Y$ on the modifier.
It helps if you use a different view, that clearly shows the axis you want to work on (like top view)
The Fix: If you don't need the non-uniform object-level scale for animation, CtrlA > Apply the scale of your object.
Why that's a fix isn't obvious .. your object is scaled equally in X and Y, (though not in Z). I would have thought normals would be transformed consistently.. it's strange that normals of adjacent faces which previously were equal, are ...
Just in case somebody comes across this thread looking for the answer (because I believe OP is no longer waiting for one): If you have a SubDivision modifier, it should come AFTER the Mirror modifier, so that the transition gets smoothed.