Sadly overlapping multiple filled objects will not result in them being cut as one part when they are sent to the laser.
While filled objects that are overlapping may look like one solid shape in 'normal' view, switching to 'outline view' will make it easy to see there are many overlapping lines present.
If you have overlapping paths you'll need to merge them, otherwise your design will end up in a whole bunch of separate pieces.
In Adobe Illustrator the 'pathfinder > unite' tool is helpful for this purpose and in Inkscape the 'path > union' tool works well.
Let's see that in action:
Illustrator: Pathfinder > Unite
Inkscape: Path > Union
*Make sure to ungroup parts before running a unite function as this often doesn't work on grouped parts. To check what paths are left by viewing in 'outline' mode before saving.