From what I understood your SVG is made of a white object (the face) on a black background. Therefore we can not easily select the black eye as a separate object.
To overcome this a very fast method would be to use Inkscape's Bucket Fill tool (Shift + F7 ). By this we can create a new object outlining the shape of the undelying object we had applied it to. Colors and stroke properties will will be of our current defaults and needs to be adapted to match the source.
For more complex objects, or in case we need to obtain parts of a (foreground) object we can Break apart the existing path to create new objects.
If you need to separate out a bunch of elements and/or don't want to disturb the placement of the elements on the drawing, I recommend the following procedure (Windows/Linux shortcuts included to help speed your workflow):
Open the Layers dialog (ctrl-shift-L)
Create a new layer called "New"
Create a new layer called "Working Space"
Create a new layer called "Old" (This one is optional if your
original is already in a layer)
Move your original to "Old". This will be your source, so we won't
alter it. Select the original, "Layer" -> "Move Selection to
Layer...", then select "Old" in the dialog.
Click on the eye icon next to layer "New" to hide it.
If layer "Old" is hidden, click on the eye icon next to "Old" to
Select your original and copy it (ctrl-C), select "Working Space" in
the layers dialog, paste in place (ctrl-alt-V)
Click on the eye icon next to layer "Old" to hide it.
Layer "Working Space" and your original should already be selected.
If not, select them.
"Path" -> "Break Apart" (ctrl-shiftl-K)
Shift-click on the element you want to separate out. This will
un-select it, leaving the rest of the original selected.
"Edit" -> "Delete" (del) to delete the rest of the original.
Move your separated element to "New". Select the element, "Layer"
-> "Move Selection to Layer...", then select "New" in the dialog
Repeat steps 7-14 for additional elements you want to separate.