Your top composition is longer than your second. AE is indeed pasting these layers (as evidenced by them appearing in the layer stack), but they're pasted at the same time in this second comp (just how copy/pasting of layers works) - which is beyond the duration of your comp.
You'd either want your second composition to be long enough to compensate, or you can bring these two layers to the current time by pressing Alt/Opt + [
(Note that this will align the start points of both layers, so you may want to trim them to match before copying.
* Personally, I like making my comps several seconds longer than I actually need them to be, to accommodate for this, potential for extending duration, etc. Using Shift + the numbers along the top of your keyboard allows you to drop numbered markers in the composition timeline, and you can jump to any of those by pressing the corresponding number. I like using 0 as my "End" marker for comps.