Exactly, what makes a person that person is the memories, there's nothing else.
Copy = revival
Btw Izumi didn't "jump through" anything, look again and you'll see he merely jumped while being sent. What gantz does seems to do is some form of transportation where he gradually sends the matter that constitutes the gantzer (or whatever is being transported) and instantly rebuilds it on the other side. In some cases the rebuilding process is made according to a past state from gantz memory (when there are injuries, broken suit...), and in others (revivals) the process is one-sided and gantz simply builds the gantzer (using matter transported from wherever).
Its just like the story of the grasshopper and the octopus. All year long the grasshopper kept burying acorns for winter while the octopus mooched off his girlfriend and watched TV. Then the winter came, and the grasshopper died, and the octopus ate all his acorns and also he got a racecar. Is any of this getting through to you?