The only similarity I see with #27113 is that both there and here a "cp" usage message is printed, because the dylib didn't get built. In #27113 we have not yet figured out why. In your case, the relevant error in your log seems to be:
dyld: shared cached file was build against a different libSystem.dylib, ignoring cache
This is a new error message to me. As in #27113 there is lots of "Skipping completed" in your log, which means this was not a clean build. It sounds almost like you began building zlib, then upgraded Mac OS X (or migrated MacPorts to a Mac with a different version of Mac OS X), then tried to continue the build; this certainly won't work. Please clean and try again. If that fails again, attach the complete main.log.
sudo port clean zlib
sudo port install zlib
#27113 is possibly a duplicate of #26224; please see also if any of the hypotheses listed there fit your case.
