Keys belong to timelines. Each timeline represents the values of a property (sometimes more than one). For example, a rotate key on the head bone. If you copy and paste that key to some other animation, it will only change that property: rotation on the head bone. That is separate from any other keys you may have.
It sounds like you want to copy/paste the entire pose. Instead you are copying the keys, which is not the entire pose if you have other keys, as explained in the last paragraph.
Also, in your screenshot showing what you copy, some timelines don't have a key on frame zero, so you are not copying the value for that timeline (which comes from the setup pose). As explained earlier in this thread, you can use Key Visible (ctrl+shift+L) to set a key for every timeline, then copy the keys.
Even when copying all the keys in one animation, when you paste, it will only set keys for what you copied. There are other properties that could be keyed in the animation where you pasted. Those other keys will be left unchanged and may affect the pose in a way you don't want.
Lastly, when a key is the first key on a timeline and it is on frame > 0, the setup pose is used before the key value and the transition from the setup pose to the key is stepped. If you don't want this, you need to set an earlier key and configure the curve.
It may help to do your test with only 1 or 2 keys in each animation so you can better understand what is happening. There are many keyable properties and you almost never want to set a key for ALL of them.
If you want the position of all the bones from one animation to another, you can select all bones in the viewport, copy, go to another animation, and paste the bone positions. This may be more like what you want, but note that it won't bring over keyed values like IK constraint mix, etc. You can use a combination of copying keys and bone positions.