<extremelybenshapirovoice>
Let's say, for the sake of argument, the universe destroys and recreates itself every Planck second, and that there's no such thing as "continuous existence". The universe advances "frame-by-frame" on a scale undetectable by modern science. I'm not suggesting this is how the universe actually works, but let's pretend it works like this. The question is, how would you feel about your life? Would you think you're "dying" countless times per moment? Furthermore, how would you differentiate this "constant recreation" universe from a "continuous existence" universe?
IMO, you wouldn't be able to. There is no functional difference in terms of human perception. See:
Meditations on First Philosophy
Star Trek teleportation works the same way. The existential crisis is only in your mind. As long as the destruction-recreation is nominally undetectable, your existence is preserved, because existence is only in the mind.
Addendum: There's no guarantee "portals" don't work by deconstructing you atom by atom and reconstructing you on the other side seamlessly. It's all fiction and fantasy.
</extremelybenshapirovoice>