The problem is that both AnimationPlayerEditorPlugin and AnimationTreeEditorPlugin handle AnimationTree. AnimationTree should be handled only by one plugin. Is there any reason why it is this way? What's the expected behavior?
CC @godotengine/animation

When I blocked AnimationPlayerEditorPlugin by code, AnimationTree itself worked fine.

So, I thought it would work when AnimationPlayerEditorPlugin::handles() func gets modified.

from
bool AnimationPlayerEditorPlugin::handles(Object *p_object) const { return p_object->is_class("AnimationPlayer") || p_object->is_class("AnimationTree") || p_object->is_class("AnimationMixer"); }
to
bool AnimationPlayerEditorPlugin::handles(Object *p_object) const { return p_object->is_class("AnimationPlayer") || p_object->is_class("AnimationMixer"); }

But it still didn't solve the issue because (AnimationTree)->is_class("AnimationMixer") returns true and AnimationPlayerEditorPlugin cuts in again.

The condition p_object->is_class("AnimationMixer") is still needed because without it AnimationPlayer gets broken.(Not playing animation)

This solves the bottom panel problem, but as I mentioned above it breaks AnimationPlayer.

However, that only happens when there is any connection in AnimationTree.

Edit: It was not just broken, the AnimationTree was dominating animation viewport.
And with an animation made right in the project(not external), it worked fine.
Is this okay to go?

The problem is that both AnimationPlayerEditorPlugin and AnimationTreeEditorPlugin handle AnimationTree. AnimationTree should be handled only by one plugin. Is there any reason why it is this way? What's the expected behavior?

@KoBeWi AnimationTree now be able to have its own AnimationLibrary in 4.2, allowing it to store and edit Animations/AnimationLibraries even without an external AnimationPlayer. Internally, when an AnimationTree is handled by the AnimationPlayerEditor, the AnimationPlayerEditor creates a dummy animation player inside.

So while it may need to be modified to give priority to the AnimationTreeEditor when selecting AnimationTree node, but there is nothing wrong with the AnimationPlayerEditor handling both the AnimationPlayer and the AnimationTree.