It's impossible to know what is at work without more information, but I wanted to offer some general information anyway.
It's important to understand what is happening when Blender selects objects. The first object to be selected isn't merely selected, but is also active. An active object is a special case of selected object, and there can only ever be one active object at a time, even if multiple objects are selected. When adding multiple objects to any selection one at a time, the last one selected is always the active object. When you use the Make Links menu, as you did above, Blender needs to know what object is going to be the source object for the data transfer. The active object is always the one from which data is transferred, and all other selected objects are the ones to which that same data is transferred.
Now, Blender also remembers the last object to have been selected, even if no objects are selected currently. This is important, because there are ways to select multiple objects without selecting them one at a time. If it were the case that only the last object selected is ever the active object, then which object would get designated as active when all objects are selected at once? The answer is, the one that Blender remembers being exclusively selected previously. And it turns out that every object is designated as exclusively selected (and therefore active) as soon as it is initially created. Meaning that there will always be an object in the scene that Blender remembers to have been exclusively selected previously. So there will never be a condition for which selection outcomes prevent Blender from knowing which object to designate as active.
It's also important to understand that objects which are currently invisible, or which are marked as unselectable currently in the Outliner, will not be added to any selection made via the All By Type menu. This rule also applies to the active object. If the active object is invisible or unselectable, it cannot be sourced in a linking operation. Performing the link with the active object cut off in this way will result in no linking being performed. Invisible or unselectable objects will, as a rule, not be involved in any operations which require selections in order to function. So check to see if you have any visibility or selectability issues.
I noticed in your screenshot that you don't have your selectability column exposed in the Outliner. So you aren't yet even in a position to see what the selectability status is for the objects in your scene. Float your cursor to the top-right of the Outliner, above your listed objects, and you will see the icon for the Restriction Toggle. The selectability toggle option will be found under there.
Until you give more info, there won't be much more anyone can say about your file.