Given what I see in your video, you are importing an animation with a different skeleton then the one of the skeleton you are trying to use.
And in fact it is.
Here is the T pose skeleton VS the Animation Tpose.
Ergo, Your re-targeting the wrong skeleton.
Also, the tutorial you linked uses TurboSquid characters. They might be rigged differently, which would explain why.
What you can do is extract the first T-pose off your FBX animation, save it as an armature and import it into Unreal.
You can then re-target the Maximo character to this T-Pose.
After that you should be 100% able to import the animation using the specifically re-targed skeleton.
Also, Blender says that the take-2-fight_HUMANIK_FW3_segment.fbx will be a 1:1 of the skeleton of the character with just basic re-targeting.
So import that animation instead of the Maximo labled one and tell me how it goes.
Oh, before you ask, you have to rename all the bones in the animation to remove “character” from in front.
You can easily do this with a blender script. Hang on. typing it up.
Run it from Script mode in Console:
for bone in bpy.data.objects’Character1_Reference’].data.bones: bone.name = bone.name[11:]
enter 2 times after it.
Now that this is done you can go and set the framerate to 25 unless you want to wait a lifetime between imports.
Export the new FBX with the following settings
-Z forward, Y up (suppresses FBX error message).
select Armat by clicking.
Move to armature tab at top of export dialog.
Disable Leaf Bones.
Select primary y, sec x.
check everything in animation tab + export.
Now in Unreal, Import settings, convert scene and add an Import rotation of 90 on X then import, and there you have it.
Note, for this I have ALL re-targeting set to skeleton on the skeleton, including hips. And here is an FBX you ca re-import.
http://mosthostla.com/downloads/animation_fix.rar